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^L$fH~fIn| b^L$$fHn{X$=@f/rf/zd $H []^A^f(fDT$f.T$ff.D$Hbf(\$f.ȇT$ff.D$H $# $XH []A^^f(fvHEH8Pf(\\z\$f.Ef(fW qkHE $H8Pf(\z $fWTaH9fL$HHNfMHMfI)H*I9HD$8LI*LOfIH*fI*L)|$f(^f(YXUD$fH*ID$Y%\YfH*Y^Xf.!f(Qf(IGfIYT$0M)XL$(\$XD$ fH*HD$8HH*YfI*^yL,fID$H*誥HD$8D$fL)HH*芥L\$L)HXfHH*\$_XD$KD&Lt$@D$fH*;d$L9|$8L$(T$0Xd$HY D$N%XXf(fTf.]T$0fDH;SH;D$SL$|$\zYD$ ^XD$f/wf/D$0s?xML,fID$M)H*aHD$8D$(fL)HH*Ad$(IFXfH*d$(XD$(HD$@JD D$(fH*XD$(T$HL$\\Y\f/s=f(T$\Yf/f(nsT$Xf/L;l$PMOM)L9MOffH*fI~DH,f5ofUH*f(fT\fVf(H\$Xf(T$ \$wT$ \$f(ff.[ff.ff.f/Ir fDHH?D$APfWrL$$~\>r$^f( wHH,f[ff.SHH@D$8f(L$ p+w;~f/D$ o|$ f/~fl$D$^XD$0,f(^{qXD$L$\f/D$s}H;SYq\$0H;f(YXXL$f(L$ ^\d$(YL$SL$f(F\Y\f/D$^H;SD$D$(qf(f/D$vfWcXT$8 f(fT}T$Xp|$T$\f/w,H@[fDH;SX\|YH@[fWH@[f p-x|ft$Yl$YXf.w[Qf(XL$f(Xf.wTQ\T$ f(f(X^f(YXXL$^L$0@f(Zt|$f(f(L$?tL$f(SHH05{D$ \f(MoD$(H;Sf/D$ D$H;SYD$(nmn{T$\f(Yf/r~f(T$\$n\$D$f(nL$^{XrL,MaT$ff.E„EH0L[f/ArAHH%.200s() keywords must be strings%s() got an unexpected keyword argument '%U' while calling a Python objectNULL result without error in PyObject_Call__int__ returned non-int (type %.200s). The ability to return an instance of a strict subclass of int is deprecated, and may be removed in a future version of Python.__%.4s__ returned non-%.4s (type %.200s)value too large to convert to int%.200s does not export expected C variable %.200sC variable %.200s.%.200s has wrong signature (expected %.500s, got %.500s)%.200s does not export expected C function %.200sC function %.200s.%.200s has wrong signature (expected %.500s, got %.500s)Interpreter change detected - this module can only be loaded into one interpreter per process.%.200s.%.200s is not a type object%.200s.%.200s size changed, may indicate binary incompatibility. Expected %zd from C header, got %zd from PyObject%s.%s size changed, may indicate binary incompatibility. Expected %zd from C header, got %zd from PyObject'%.200s' object is unsliceable%s() got multiple values for keyword argument '%U'calling %R should have returned an instance of BaseException, not %Rraise: exception class must be a subclass of BaseExceptioninvalid vtable found for imported typeCannot convert %.200s to %.200s'%.200s' object has no attribute '%U'numpy/random/mtrand.cpython-39-x86_64-linux-gnu.so.p/numpy/random/mtrand.pyx.c%.200s() takes %.8s %zd positional argument%.1s (%zd given)numpy.random.mtrand.RandomState.randomnumpy.random.mtrand.RandomState.__getstate__numpy.random.mtrand.RandomState.__str__numpy.random.mtrand.RandomState._initialize_bit_generatorcannot fit '%.200s' into an index-sized integer'%.200s' object is not subscriptabletoo many values to unpack (expected %zd)need more than %zd value%.1s to unpacknumpy.random.mtrand.RandomState.__reduce__numpy.random.mtrand.RandomState.noncentral_chisquarenumpy.random.mtrand.RandomState.fnumpy.random.mtrand.RandomState.waldnumpy.random.mtrand.RandomState.randnumpy.random.mtrand.RandomState.randnnumpy.random.mtrand.RandomState.__repr__numpy.random.mtrand.RandomState.paretonumpy.random.mtrand.RandomState.weibullnumpy.random.mtrand.RandomState.chisquarenumpy.random.mtrand.RandomState.standard_gammanumpy.random.mtrand.RandomState.standard_tnumpy.random.mtrand.RandomState.powernumpy.random.mtrand.RandomState.normalnumpy.random.mtrand.RandomState.logisticnumpy.random.mtrand.RandomState.lognormalnumpy.random.mtrand.RandomState.laplacenumpy.random.mtrand.RandomState.gumbelnumpy.random.mtrand.RandomState.gammanumpy.random.mtrand.RandomState.__init__hasattr(): attribute name must be stringnumpy.random.mtrand.int64_to_longnumpy.random.mtrand.RandomState.logseriesnumpy.random.mtrand.RandomState.geometricnumpy.random.mtrand.RandomState.zipfnumpy.random.mtrand.RandomState.negative_binomialnumpy.random.mtrand.RandomState.triangularnumpy.random.mtrand.RandomState.uniformnumpy.random.mtrand.RandomState.randintModule 'mtrand' has already been imported. Re-initialisation is not supported.compile time version %s of module '%.100s' does not match runtime version %sbase class '%.200s' is not a heap typeextension type '%.200s' has no __dict__ slot, but base type '%.200s' has: either add 'cdef dict __dict__' to the extension type or add '__slots__ = [...]' to the base typemultiple bases have vtable conflict: '%.200s' and '%.200s'numpy.random._bounded_integersPyObject *(PyObject *, PyObject *, PyObject *, int, int, bitgen_t *, PyObject *)int (double, PyObject *, __pyx_t_5numpy_6random_7_common_constraint_type)int (PyArrayObject *, PyObject *, __pyx_t_5numpy_6random_7_common_constraint_type)PyObject *(void *, bitgen_t *, PyObject *, PyObject *, PyObject *)PyObject *(PyObject *, PyArrayObject *)PyObject *(void *, void *, PyObject *, PyObject *, int, PyObject *, PyObject *, __pyx_t_5numpy_6random_7_common_constraint_type, PyObject *, PyObject *, __pyx_t_5numpy_6random_7_common_constraint_type, PyObject *, PyObject *, __pyx_t_5numpy_6random_7_common_constraint_type, PyObject *)PyObject *(void *, void *, PyObject *, PyObject *, int, int, PyObject *, PyObject *, __pyx_t_5numpy_6random_7_common_constraint_type, PyObject *, PyObject *, __pyx_t_5numpy_6random_7_common_constraint_type, PyObject *, PyObject *, __pyx_t_5numpy_6random_7_common_constraint_type)PyObject *(void *, void *, PyObject *, PyObject *, PyArrayObject *, PyObject *, __pyx_t_5numpy_6random_7_common_constraint_type, PyArrayObject *, PyObject *, __pyx_t_5numpy_6random_7_common_constraint_type, PyArrayObject *, PyObject *, __pyx_t_5numpy_6random_7_common_constraint_type)_ARRAY_API is not PyCapsule objectmodule compiled against ABI version 0x%x but this version of numpy is 0x%xmodule compiled against API version 0x%x but this version of numpy is 0x%x . Check the section C-API incompatibility at the Troubleshooting ImportError section at https://numpy.org/devdocs/user/troubleshooting-importerror.html#c-api-incompatibility for indications on how to solve this problem .FATAL: module compiled as unknown endianFATAL: module compiled as little endian, but detected different endianness at runtimenumpy.random.mtrand.get_bit_generatornumpy.random.mtrand.set_bit_generatornumpy.random.mtrand.RandomState.rayleighnumpy.random.mtrand.RandomState.exponentialnumpy.random.mtrand.RandomState.set_statenumpy.random.mtrand.RandomState.shufflejoin() result is too long for a Python stringnumpy.random.mtrand.RandomState.get_statenumpy.random.mtrand.RandomState.seednumpy.random.mtrand.RandomState.poissonnumpy.random.mtrand.RandomState.noncentral_fnumpy.random.mtrand.RandomState.bytesnumpy.random.mtrand.RandomState.hypergeometricnumpy.random.mtrand.RandomState.random_samplenumpy.random.mtrand.RandomState.dirichletnumpy.random.mtrand.RandomState.standard_exponentialnumpy.random.mtrand.RandomState.standard_normalnumpy.random.mtrand.RandomState.standard_cauchynumpy.random.mtrand.RandomState.tomaxintnumpy.random.mtrand.RandomState.random_integersnumpy.random.mtrand.RandomState.__setstate__numpy.random.mtrand.RandomState.multinomialnumpy.random.mtrand.RandomState.betanumpy.random.mtrand.RandomState.vonmisesnumpy.random.mtrand.RandomState.binomialnumpy.random.mtrand.RandomState.multivariate_normalnumpy.random.mtrand.RandomState.permutationnumpy.random.mtrand.RandomState.choice'%.200s' object does not support slice %.10snumpy.random.mtrand.RandomState RandomState(seed=None) Container for the slow Mersenne Twister pseudo-random number generator. Consider using a different BitGenerator with the Generator container instead. `RandomState` and `Generator` expose a number of methods for generating random numbers drawn from a variety of probability distributions. In addition to the distribution-specific arguments, each method takes a keyword argument `size` that defaults to ``None``. If `size` is ``None``, then a single value is generated and returned. If `size` is an integer, then a 1-D array filled with generated values is returned. If `size` is a tuple, then an array with that shape is filled and returned. **Compatibility Guarantee** A fixed bit generator using a fixed seed and a fixed series of calls to 'RandomState' methods using the same parameters will always produce the same results up to roundoff error except when the values were incorrect. `RandomState` is effectively frozen and will only receive updates that are required by changes in the internals of Numpy. More substantial changes, including algorithmic improvements, are reserved for `Generator`. Parameters ---------- seed : {None, int, array_like, BitGenerator}, optional Random seed used to initialize the pseudo-random number generator or an instantized BitGenerator. If an integer or array, used as a seed for the MT19937 BitGenerator. Values can be any integer between 0 and 2**32 - 1 inclusive, an array (or other sequence) of such integers, or ``None`` (the default). If `seed` is ``None``, then the `MT19937` BitGenerator is initialized by reading data from ``/dev/urandom`` (or the Windows analogue) if available or seed from the clock otherwise. Notes ----- The Python stdlib module "random" also contains a Mersenne Twister pseudo-random number generator with a number of methods that are similar to the ones available in `RandomState`. `RandomState`, besides being NumPy-aware, has the advantage that it provides a much larger number of probability distributions to choose from. See Also -------- Generator MT19937 numpy.random.BitGenerator an integer is required__pyx_capi__name__loader__loader__file__origin__package__parent__path__submodule_search_locationsMissing type objectcannot import name %S%s (%s:%d)at leastat mostrandomnumpy/random/mtrand.pyxexactly__getstate__BitGenerator__reduce__noncentral_chisquarewaldrandnparetoweibullstandard_gammastandard_tpowerlogisticlognormallaplacegumbelname '%U' is not defined__init__logserieszipfnegative_binomialtriangularuniformrandintbuiltinscython_runtime__builtins__%d.%d4294967296complexnumpydtypeflatiterbroadcastndarraygenericnumberunsignedintegerinexactcomplexfloatingflexiblecharacterufuncnumpy.random.bit_generatorSeedSequenceSeedlessSequencenumpy.random._commondoubleLEGACY_POISSON_LAM_MAXuint64_tMAXSIZE_rand_uint64_rand_uint32_rand_uint16_rand_uint8_rand_bool_rand_int64_rand_int32_rand_int16_rand_int8check_constraintcheck_array_constraintdouble (double *, npy_intp)kahan_sumdouble_fillvalidate_output_shapecontdisccont_broadcast_3discrete_broadcast_iiinumpy.core._multiarray_umath_ARRAY_API_ARRAY_API not found_ARRAY_API is NULL pointernumpy/__init__.cython-30.pxdnumpy.import_arrayinit numpy.random.mtrandnumpy.random.mtrand.ranfnumpy.random.mtrand.sampleset_bit_generatorrayleighset_stateshuffleget_statenumpy.random.mtrand.seedpoissonnoncentral_fbyteshypergeometricrandom_sampledirichletstandard_exponentialstandard_normalstandard_cauchytomaxintrandom_integers__setstate__multinomialbetavonmisesnumpy.PyArray_MultiIterNew2numpy.PyArray_MultiIterNew3multivariate_normalpermutationassignmentdeletionchoice__repr__get_bit_generatorx0`0&&&gg^^^!!BB877. (''''l.(Z(((D, This is an alias of `random_sample`. See `random_sample` for the complete documentation. This is an alias of `random_sample`. See `random_sample` for the complete documentation. Sets the singleton RandomState's bit generator Parameters ---------- bitgen A bit generator instance Notes ----- The singleton RandomState provides the random variate generators in the ``numpy.random``namespace. This function, and its counterpart get method, provides a path to hot-swap the default MT19937 bit generator with a user provided alternative. These function are intended to provide a continuous path where a single underlying bit generator can be used both with an instance of ``Generator`` and with the singleton instance of RandomState. See Also -------- get_bit_generator numpy.random.Generator Returns the singleton RandomState's bit generator Returns ------- BitGenerator The bit generator that underlies the singleton RandomState instance Notes ----- The singleton RandomState provides the random variate generators in the ``numpy.random`` namespace. This function, and its counterpart set method, provides a path to hot-swap the default MT19937 bit generator with a user provided alternative. These function are intended to provide a continuous path where a single underlying bit generator can be used both with an instance of ``Generator`` and with the singleton instance of RandomState. See Also -------- set_bit_generator numpy.random.Generator seed(seed=None) Reseed the singleton RandomState instance. Notes ----- This is a convenience, legacy function that exists to support older code that uses the singleton RandomState. Best practice is to use a dedicated ``Generator`` instance rather than the random variate generation methods exposed directly in the random module. See Also -------- numpy.random.Generator permutation(x) Randomly permute a sequence, or return a permuted range. If `x` is a multi-dimensional array, it is only shuffled along its first index. .. note:: New code should use the `~numpy.random.Generator.permutation` method of a `~numpy.random.Generator` instance instead; please see the :ref:`random-quick-start`. Parameters ---------- x : int or array_like If `x` is an integer, randomly permute ``np.arange(x)``. If `x` is an array, make a copy and shuffle the elements randomly. Returns ------- out : ndarray Permuted sequence or array range. See Also -------- random.Generator.permutation: which should be used for new code. Examples -------- >>> np.random.permutation(10) array([1, 7, 4, 3, 0, 9, 2, 5, 8, 6]) # random >>> np.random.permutation([1, 4, 9, 12, 15]) array([15, 1, 9, 4, 12]) # random >>> arr = np.arange(9).reshape((3, 3)) >>> np.random.permutation(arr) array([[6, 7, 8], # random [0, 1, 2], [3, 4, 5]]) shuffle(x) Modify a sequence in-place by shuffling its contents. This function only shuffles the array along the first axis of a multi-dimensional array. The order of sub-arrays is changed but their contents remains the same. .. note:: New code should use the `~numpy.random.Generator.shuffle` method of a `~numpy.random.Generator` instance instead; please see the :ref:`random-quick-start`. Parameters ---------- x : ndarray or MutableSequence The array, list or mutable sequence to be shuffled. Returns ------- None See Also -------- random.Generator.shuffle: which should be used for new code. Examples -------- >>> arr = np.arange(10) >>> np.random.shuffle(arr) >>> arr [1 7 5 2 9 4 3 6 0 8] # random Multi-dimensional arrays are only shuffled along the first axis: >>> arr = np.arange(9).reshape((3, 3)) >>> np.random.shuffle(arr) >>> arr array([[3, 4, 5], # random [6, 7, 8], [0, 1, 2]]) dirichlet(alpha, size=None) Draw samples from the Dirichlet distribution. Draw `size` samples of dimension k from a Dirichlet distribution. A Dirichlet-distributed random variable can be seen as a multivariate generalization of a Beta distribution. The Dirichlet distribution is a conjugate prior of a multinomial distribution in Bayesian inference. .. note:: New code should use the `~numpy.random.Generator.dirichlet` method of a `~numpy.random.Generator` instance instead; please see the :ref:`random-quick-start`. Parameters ---------- alpha : sequence of floats, length k Parameter of the distribution (length ``k`` for sample of length ``k``). size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n)``, then ``m * n * k`` samples are drawn. Default is None, in which case a vector of length ``k`` is returned. Returns ------- samples : ndarray, The drawn samples, of shape ``(size, k)``. Raises ------ ValueError If any value in ``alpha`` is less than or equal to zero See Also -------- random.Generator.dirichlet: which should be used for new code. Notes ----- The Dirichlet distribution is a distribution over vectors :math:`x` that fulfil the conditions :math:`x_i>0` and :math:`\sum_{i=1}^k x_i = 1`. The probability density function :math:`p` of a Dirichlet-distributed random vector :math:`X` is proportional to .. math:: p(x) \propto \prod_{i=1}^{k}{x^{\alpha_i-1}_i}, where :math:`\alpha` is a vector containing the positive concentration parameters. The method uses the following property for computation: let :math:`Y` be a random vector which has components that follow a standard gamma distribution, then :math:`X = \frac{1}{\sum_{i=1}^k{Y_i}} Y` is Dirichlet-distributed References ---------- .. [1] David McKay, "Information Theory, Inference and Learning Algorithms," chapter 23, http://www.inference.org.uk/mackay/itila/ .. [2] Wikipedia, "Dirichlet distribution", https://en.wikipedia.org/wiki/Dirichlet_distribution Examples -------- Taking an example cited in Wikipedia, this distribution can be used if one wanted to cut strings (each of initial length 1.0) into K pieces with different lengths, where each piece had, on average, a designated average length, but allowing some variation in the relative sizes of the pieces. >>> s = np.random.dirichlet((10, 5, 3), 20).transpose() >>> import matplotlib.pyplot as plt >>> plt.barh(range(20), s[0]) >>> plt.barh(range(20), s[1], left=s[0], color='g') >>> plt.barh(range(20), s[2], left=s[0]+s[1], color='r') >>> plt.title("Lengths of Strings") multinomial(n, pvals, size=None) Draw samples from a multinomial distribution. The multinomial distribution is a multivariate generalization of the binomial distribution. Take an experiment with one of ``p`` possible outcomes. An example of such an experiment is throwing a dice, where the outcome can be 1 through 6. Each sample drawn from the distribution represents `n` such experiments. Its values, ``X_i = [X_0, X_1, ..., X_p]``, represent the number of times the outcome was ``i``. .. note:: New code should use the `~numpy.random.Generator.multinomial` method of a `~numpy.random.Generator` instance instead; please see the :ref:`random-quick-start`. Parameters ---------- n : int Number of experiments. pvals : sequence of floats, length p Probabilities of each of the ``p`` different outcomes. These must sum to 1 (however, the last element is always assumed to account for the remaining probability, as long as ``sum(pvals[:-1]) <= 1)``. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. Default is None, in which case a single value is returned. Returns ------- out : ndarray The drawn samples, of shape *size*, if that was provided. If not, the shape is ``(N,)``. In other words, each entry ``out[i,j,...,:]`` is an N-dimensional value drawn from the distribution. See Also -------- random.Generator.multinomial: which should be used for new code. Examples -------- Throw a dice 20 times: >>> np.random.multinomial(20, [1/6.]*6, size=1) array([[4, 1, 7, 5, 2, 1]]) # random It landed 4 times on 1, once on 2, etc. Now, throw the dice 20 times, and 20 times again: >>> np.random.multinomial(20, [1/6.]*6, size=2) array([[3, 4, 3, 3, 4, 3], # random [2, 4, 3, 4, 0, 7]]) For the first run, we threw 3 times 1, 4 times 2, etc. For the second, we threw 2 times 1, 4 times 2, etc. A loaded die is more likely to land on number 6: >>> np.random.multinomial(100, [1/7.]*5 + [2/7.]) array([11, 16, 14, 17, 16, 26]) # random The probability inputs should be normalized. As an implementation detail, the value of the last entry is ignored and assumed to take up any leftover probability mass, but this should not be relied on. A biased coin which has twice as much weight on one side as on the other should be sampled like so: >>> np.random.multinomial(100, [1.0 / 3, 2.0 / 3]) # RIGHT array([38, 62]) # random not like: >>> np.random.multinomial(100, [1.0, 2.0]) # WRONG Traceback (most recent call last): ValueError: pvals < 0, pvals > 1 or pvals contains NaNs multivariate_normal(mean, cov, size=None, check_valid='warn', tol=1e-8) Draw random samples from a multivariate normal distribution. The multivariate normal, multinormal or Gaussian distribution is a generalization of the one-dimensional normal distribution to higher dimensions. Such a distribution is specified by its mean and covariance matrix. These parameters are analogous to the mean (average or "center") and variance (standard deviation, or "width," squared) of the one-dimensional normal distribution. .. note:: New code should use the `~numpy.random.Generator.multivariate_normal` method of a `~numpy.random.Generator` instance instead; please see the :ref:`random-quick-start`. Parameters ---------- mean : 1-D array_like, of length N Mean of the N-dimensional distribution. cov : 2-D array_like, of shape (N, N) Covariance matrix of the distribution. It must be symmetric and positive-semidefinite for proper sampling. size : int or tuple of ints, optional Given a shape of, for example, ``(m,n,k)``, ``m*n*k`` samples are generated, and packed in an `m`-by-`n`-by-`k` arrangement. Because each sample is `N`-dimensional, the output shape is ``(m,n,k,N)``. If no shape is specified, a single (`N`-D) sample is returned. check_valid : { 'warn', 'raise', 'ignore' }, optional Behavior when the covariance matrix is not positive semidefinite. tol : float, optional Tolerance when checking the singular values in covariance matrix. cov is cast to double before the check. Returns ------- out : ndarray The drawn samples, of shape *size*, if that was provided. If not, the shape is ``(N,)``. In other words, each entry ``out[i,j,...,:]`` is an N-dimensional value drawn from the distribution. See Also -------- random.Generator.multivariate_normal: which should be used for new code. Notes ----- The mean is a coordinate in N-dimensional space, which represents the location where samples are most likely to be generated. This is analogous to the peak of the bell curve for the one-dimensional or univariate normal distribution. Covariance indicates the level to which two variables vary together. From the multivariate normal distribution, we draw N-dimensional samples, :math:`X = [x_1, x_2, ... x_N]`. The covariance matrix element :math:`C_{ij}` is the covariance of :math:`x_i` and :math:`x_j`. The element :math:`C_{ii}` is the variance of :math:`x_i` (i.e. its "spread"). Instead of specifying the full covariance matrix, popular approximations include: - Spherical covariance (`cov` is a multiple of the identity matrix) - Diagonal covariance (`cov` has non-negative elements, and only on the diagonal) This geometrical property can be seen in two dimensions by plotting generated data-points: >>> mean = [0, 0] >>> cov = [[1, 0], [0, 100]] # diagonal covariance Diagonal covariance means that points are oriented along x or y-axis: >>> import matplotlib.pyplot as plt >>> x, y = np.random.multivariate_normal(mean, cov, 5000).T >>> plt.plot(x, y, 'x') >>> plt.axis('equal') >>> plt.show() Note that the covariance matrix must be positive semidefinite (a.k.a. nonnegative-definite). Otherwise, the behavior of this method is undefined and backwards compatibility is not guaranteed. References ---------- .. [1] Papoulis, A., "Probability, Random Variables, and Stochastic Processes," 3rd ed., New York: McGraw-Hill, 1991. .. [2] Duda, R. O., Hart, P. E., and Stork, D. G., "Pattern Classification," 2nd ed., New York: Wiley, 2001. Examples -------- >>> mean = (1, 2) >>> cov = [[1, 0], [0, 1]] >>> x = np.random.multivariate_normal(mean, cov, (3, 3)) >>> x.shape (3, 3, 2) Here we generate 800 samples from the bivariate normal distribution with mean [0, 0] and covariance matrix [[6, -3], [-3, 3.5]]. The expected variances of the first and second components of the sample are 6 and 3.5, respectively, and the expected correlation coefficient is -3/sqrt(6*3.5) ≈ -0.65465. >>> cov = np.array([[6, -3], [-3, 3.5]]) >>> pts = np.random.multivariate_normal([0, 0], cov, size=800) Check that the mean, covariance, and correlation coefficient of the sample are close to the expected values: >>> pts.mean(axis=0) array([ 0.0326911 , -0.01280782]) # may vary >>> np.cov(pts.T) array([[ 5.96202397, -2.85602287], [-2.85602287, 3.47613949]]) # may vary >>> np.corrcoef(pts.T)[0, 1] -0.6273591314603949 # may vary We can visualize this data with a scatter plot. The orientation of the point cloud illustrates the negative correlation of the components of this sample. >>> import matplotlib.pyplot as plt >>> plt.plot(pts[:, 0], pts[:, 1], '.', alpha=0.5) >>> plt.axis('equal') >>> plt.grid() >>> plt.show() logseries(p, size=None) Draw samples from a logarithmic series distribution. Samples are drawn from a log series distribution with specified shape parameter, 0 <= ``p`` < 1. .. note:: New code should use the `~numpy.random.Generator.logseries` method of a `~numpy.random.Generator` instance instead; please see the :ref:`random-quick-start`. Parameters ---------- p : float or array_like of floats Shape parameter for the distribution. Must be in the range [0, 1). size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``p`` is a scalar. Otherwise, ``np.array(p).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized logarithmic series distribution. See Also -------- scipy.stats.logser : probability density function, distribution or cumulative density function, etc. random.Generator.logseries: which should be used for new code. Notes ----- The probability density for the Log Series distribution is .. math:: P(k) = \frac{-p^k}{k \ln(1-p)}, where p = probability. The log series distribution is frequently used to represent species richness and occurrence, first proposed by Fisher, Corbet, and Williams in 1943 [2]. It may also be used to model the numbers of occupants seen in cars [3]. References ---------- .. [1] Buzas, Martin A.; Culver, Stephen J., Understanding regional species diversity through the log series distribution of occurrences: BIODIVERSITY RESEARCH Diversity & Distributions, Volume 5, Number 5, September 1999 , pp. 187-195(9). .. [2] Fisher, R.A,, A.S. Corbet, and C.B. Williams. 1943. The relation between the number of species and the number of individuals in a random sample of an animal population. Journal of Animal Ecology, 12:42-58. .. [3] D. J. Hand, F. Daly, D. Lunn, E. Ostrowski, A Handbook of Small Data Sets, CRC Press, 1994. .. [4] Wikipedia, "Logarithmic distribution", https://en.wikipedia.org/wiki/Logarithmic_distribution Examples -------- Draw samples from the distribution: >>> a = .6 >>> s = np.random.logseries(a, 10000) >>> import matplotlib.pyplot as plt >>> count, bins, ignored = plt.hist(s) # plot against distribution >>> def logseries(k, p): ... return -p**k/(k*np.log(1-p)) >>> plt.plot(bins, logseries(bins, a)*count.max()/ ... logseries(bins, a).max(), 'r') >>> plt.show() hypergeometric(ngood, nbad, nsample, size=None) Draw samples from a Hypergeometric distribution. Samples are drawn from a hypergeometric distribution with specified parameters, `ngood` (ways to make a good selection), `nbad` (ways to make a bad selection), and `nsample` (number of items sampled, which is less than or equal to the sum ``ngood + nbad``). .. note:: New code should use the `~numpy.random.Generator.hypergeometric` method of a `~numpy.random.Generator` instance instead; please see the :ref:`random-quick-start`. Parameters ---------- ngood : int or array_like of ints Number of ways to make a good selection. Must be nonnegative. nbad : int or array_like of ints Number of ways to make a bad selection. Must be nonnegative. nsample : int or array_like of ints Number of items sampled. Must be at least 1 and at most ``ngood + nbad``. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if `ngood`, `nbad`, and `nsample` are all scalars. Otherwise, ``np.broadcast(ngood, nbad, nsample).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized hypergeometric distribution. Each sample is the number of good items within a randomly selected subset of size `nsample` taken from a set of `ngood` good items and `nbad` bad items. See Also -------- scipy.stats.hypergeom : probability density function, distribution or cumulative density function, etc. random.Generator.hypergeometric: which should be used for new code. Notes ----- The probability density for the Hypergeometric distribution is .. math:: P(x) = \frac{\binom{g}{x}\binom{b}{n-x}}{\binom{g+b}{n}}, where :math:`0 \le x \le n` and :math:`n-b \le x \le g` for P(x) the probability of ``x`` good results in the drawn sample, g = `ngood`, b = `nbad`, and n = `nsample`. Consider an urn with black and white marbles in it, `ngood` of them are black and `nbad` are white. If you draw `nsample` balls without replacement, then the hypergeometric distribution describes the distribution of black balls in the drawn sample. Note that this distribution is very similar to the binomial distribution, except that in this case, samples are drawn without replacement, whereas in the Binomial case samples are drawn with replacement (or the sample space is infinite). As the sample space becomes large, this distribution approaches the binomial. References ---------- .. [1] Lentner, Marvin, "Elementary Applied Statistics", Bogden and Quigley, 1972. .. [2] Weisstein, Eric W. "Hypergeometric Distribution." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/HypergeometricDistribution.html .. [3] Wikipedia, "Hypergeometric distribution", https://en.wikipedia.org/wiki/Hypergeometric_distribution Examples -------- Draw samples from the distribution: >>> ngood, nbad, nsamp = 100, 2, 10 # number of good, number of bad, and number of samples >>> s = np.random.hypergeometric(ngood, nbad, nsamp, 1000) >>> from matplotlib.pyplot import hist >>> hist(s) # note that it is very unlikely to grab both bad items Suppose you have an urn with 15 white and 15 black marbles. If you pull 15 marbles at random, how likely is it that 12 or more of them are one color? >>> s = np.random.hypergeometric(15, 15, 15, 100000) >>> sum(s>=12)/100000. + sum(s<=3)/100000. # answer = 0.003 ... pretty unlikely! geometric(p, size=None) Draw samples from the geometric distribution. Bernoulli trials are experiments with one of two outcomes: success or failure (an example of such an experiment is flipping a coin). The geometric distribution models the number of trials that must be run in order to achieve success. It is therefore supported on the positive integers, ``k = 1, 2, ...``. The probability mass function of the geometric distribution is .. math:: f(k) = (1 - p)^{k - 1} p where `p` is the probability of success of an individual trial. .. note:: New code should use the `~numpy.random.Generator.geometric` method of a `~numpy.random.Generator` instance instead; please see the :ref:`random-quick-start`. Parameters ---------- p : float or array_like of floats The probability of success of an individual trial. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``p`` is a scalar. Otherwise, ``np.array(p).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized geometric distribution. See Also -------- random.Generator.geometric: which should be used for new code. Examples -------- Draw ten thousand values from the geometric distribution, with the probability of an individual success equal to 0.35: >>> z = np.random.geometric(p=0.35, size=10000) How many trials succeeded after a single run? >>> (z == 1).sum() / 10000. 0.34889999999999999 #random zipf(a, size=None) Draw samples from a Zipf distribution. Samples are drawn from a Zipf distribution with specified parameter `a` > 1. The Zipf distribution (also known as the zeta distribution) is a discrete probability distribution that satisfies Zipf's law: the frequency of an item is inversely proportional to its rank in a frequency table. .. note:: New code should use the `~numpy.random.Generator.zipf` method of a `~numpy.random.Generator` instance instead; please see the :ref:`random-quick-start`. Parameters ---------- a : float or array_like of floats Distribution parameter. Must be greater than 1. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``a`` is a scalar. Otherwise, ``np.array(a).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized Zipf distribution. See Also -------- scipy.stats.zipf : probability density function, distribution, or cumulative density function, etc. random.Generator.zipf: which should be used for new code. Notes ----- The probability density for the Zipf distribution is .. math:: p(k) = \frac{k^{-a}}{\zeta(a)}, for integers :math:`k \geq 1`, where :math:`\zeta` is the Riemann Zeta function. It is named for the American linguist George Kingsley Zipf, who noted that the frequency of any word in a sample of a language is inversely proportional to its rank in the frequency table. References ---------- .. [1] Zipf, G. K., "Selected Studies of the Principle of Relative Frequency in Language," Cambridge, MA: Harvard Univ. Press, 1932. Examples -------- Draw samples from the distribution: >>> a = 4.0 >>> n = 20000 >>> s = np.random.zipf(a, n) Display the histogram of the samples, along with the expected histogram based on the probability density function: >>> import matplotlib.pyplot as plt >>> from scipy.special import zeta # doctest: +SKIP `bincount` provides a fast histogram for small integers. >>> count = np.bincount(s) >>> k = np.arange(1, s.max() + 1) >>> plt.bar(k, count[1:], alpha=0.5, label='sample count') >>> plt.plot(k, n*(k**-a)/zeta(a), 'k.-', alpha=0.5, ... label='expected count') # doctest: +SKIP >>> plt.semilogy() >>> plt.grid(alpha=0.4) >>> plt.legend() >>> plt.title(f'Zipf sample, a={a}, size={n}') >>> plt.show() poisson(lam=1.0, size=None) Draw samples from a Poisson distribution. The Poisson distribution is the limit of the binomial distribution for large N. .. note:: New code should use the `~numpy.random.Generator.poisson` method of a `~numpy.random.Generator` instance instead; please see the :ref:`random-quick-start`. Parameters ---------- lam : float or array_like of floats Expected number of events occurring in a fixed-time interval, must be >= 0. A sequence must be broadcastable over the requested size. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``lam`` is a scalar. Otherwise, ``np.array(lam).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized Poisson distribution. See Also -------- random.Generator.poisson: which should be used for new code. Notes ----- The Poisson distribution .. math:: f(k; \lambda)=\frac{\lambda^k e^{-\lambda}}{k!} For events with an expected separation :math:`\lambda` the Poisson distribution :math:`f(k; \lambda)` describes the probability of :math:`k` events occurring within the observed interval :math:`\lambda`. Because the output is limited to the range of the C int64 type, a ValueError is raised when `lam` is within 10 sigma of the maximum representable value. References ---------- .. [1] Weisstein, Eric W. "Poisson Distribution." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/PoissonDistribution.html .. [2] Wikipedia, "Poisson distribution", https://en.wikipedia.org/wiki/Poisson_distribution Examples -------- Draw samples from the distribution: >>> import numpy as np >>> s = np.random.poisson(5, 10000) Display histogram of the sample: >>> import matplotlib.pyplot as plt >>> count, bins, ignored = plt.hist(s, 14, density=True) >>> plt.show() Draw each 100 values for lambda 100 and 500: >>> s = np.random.poisson(lam=(100., 500.), size=(100, 2)) negative_binomial(n, p, size=None) Draw samples from a negative binomial distribution. Samples are drawn from a negative binomial distribution with specified parameters, `n` successes and `p` probability of success where `n` is > 0 and `p` is in the interval [0, 1]. .. note:: New code should use the `~numpy.random.Generator.negative_binomial` method of a `~numpy.random.Generator` instance instead; please see the :ref:`random-quick-start`. Parameters ---------- n : float or array_like of floats Parameter of the distribution, > 0. p : float or array_like of floats Parameter of the distribution, >= 0 and <=1. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``n`` and ``p`` are both scalars. Otherwise, ``np.broadcast(n, p).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized negative binomial distribution, where each sample is equal to N, the number of failures that occurred before a total of n successes was reached. See Also -------- random.Generator.negative_binomial: which should be used for new code. Notes ----- The probability mass function of the negative binomial distribution is .. math:: P(N;n,p) = \frac{\Gamma(N+n)}{N!\Gamma(n)}p^{n}(1-p)^{N}, where :math:`n` is the number of successes, :math:`p` is the probability of success, :math:`N+n` is the number of trials, and :math:`\Gamma` is the gamma function. When :math:`n` is an integer, :math:`\frac{\Gamma(N+n)}{N!\Gamma(n)} = \binom{N+n-1}{N}`, which is the more common form of this term in the pmf. The negative binomial distribution gives the probability of N failures given n successes, with a success on the last trial. If one throws a die repeatedly until the third time a "1" appears, then the probability distribution of the number of non-"1"s that appear before the third "1" is a negative binomial distribution. References ---------- .. [1] Weisstein, Eric W. "Negative Binomial Distribution." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/NegativeBinomialDistribution.html .. [2] Wikipedia, "Negative binomial distribution", https://en.wikipedia.org/wiki/Negative_binomial_distribution Examples -------- Draw samples from the distribution: A real world example. A company drills wild-cat oil exploration wells, each with an estimated probability of success of 0.1. What is the probability of having one success for each successive well, that is what is the probability of a single success after drilling 5 wells, after 6 wells, etc.? >>> s = np.random.negative_binomial(1, 0.1, 100000) >>> for i in range(1, 11): # doctest: +SKIP ... probability = sum(s>> n, p = 10, .5 # number of trials, probability of each trial >>> s = np.random.binomial(n, p, 1000) # result of flipping a coin 10 times, tested 1000 times. A real world example. A company drills 9 wild-cat oil exploration wells, each with an estimated probability of success of 0.1. All nine wells fail. What is the probability of that happening? Let's do 20,000 trials of the model, and count the number that generate zero positive results. >>> sum(np.random.binomial(9, 0.1, 20000) == 0)/20000. # answer = 0.38885, or 38%. triangular(left, mode, right, size=None) Draw samples from the triangular distribution over the interval ``[left, right]``. The triangular distribution is a continuous probability distribution with lower limit left, peak at mode, and upper limit right. Unlike the other distributions, these parameters directly define the shape of the pdf. .. note:: New code should use the `~numpy.random.Generator.triangular` method of a `~numpy.random.Generator` instance instead; please see the :ref:`random-quick-start`. Parameters ---------- left : float or array_like of floats Lower limit. mode : float or array_like of floats The value where the peak of the distribution occurs. The value must fulfill the condition ``left <= mode <= right``. right : float or array_like of floats Upper limit, must be larger than `left`. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``left``, ``mode``, and ``right`` are all scalars. Otherwise, ``np.broadcast(left, mode, right).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized triangular distribution. See Also -------- random.Generator.triangular: which should be used for new code. Notes ----- The probability density function for the triangular distribution is .. math:: P(x;l, m, r) = \begin{cases} \frac{2(x-l)}{(r-l)(m-l)}& \text{for $l \leq x \leq m$},\\ \frac{2(r-x)}{(r-l)(r-m)}& \text{for $m \leq x \leq r$},\\ 0& \text{otherwise}. \end{cases} The triangular distribution is often used in ill-defined problems where the underlying distribution is not known, but some knowledge of the limits and mode exists. Often it is used in simulations. References ---------- .. [1] Wikipedia, "Triangular distribution" https://en.wikipedia.org/wiki/Triangular_distribution Examples -------- Draw values from the distribution and plot the histogram: >>> import matplotlib.pyplot as plt >>> h = plt.hist(np.random.triangular(-3, 0, 8, 100000), bins=200, ... density=True) >>> plt.show() wald(mean, scale, size=None) Draw samples from a Wald, or inverse Gaussian, distribution. As the scale approaches infinity, the distribution becomes more like a Gaussian. Some references claim that the Wald is an inverse Gaussian with mean equal to 1, but this is by no means universal. The inverse Gaussian distribution was first studied in relationship to Brownian motion. In 1956 M.C.K. Tweedie used the name inverse Gaussian because there is an inverse relationship between the time to cover a unit distance and distance covered in unit time. .. note:: New code should use the `~numpy.random.Generator.wald` method of a `~numpy.random.Generator` instance instead; please see the :ref:`random-quick-start`. Parameters ---------- mean : float or array_like of floats Distribution mean, must be > 0. scale : float or array_like of floats Scale parameter, must be > 0. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``mean`` and ``scale`` are both scalars. Otherwise, ``np.broadcast(mean, scale).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized Wald distribution. See Also -------- random.Generator.wald: which should be used for new code. Notes ----- The probability density function for the Wald distribution is .. math:: P(x;mean,scale) = \sqrt{\frac{scale}{2\pi x^3}}e^ \frac{-scale(x-mean)^2}{2\cdotp mean^2x} As noted above the inverse Gaussian distribution first arise from attempts to model Brownian motion. It is also a competitor to the Weibull for use in reliability modeling and modeling stock returns and interest rate processes. References ---------- .. [1] Brighton Webs Ltd., Wald Distribution, https://web.archive.org/web/20090423014010/http://www.brighton-webs.co.uk:80/distributions/wald.asp .. [2] Chhikara, Raj S., and Folks, J. Leroy, "The Inverse Gaussian Distribution: Theory : Methodology, and Applications", CRC Press, 1988. .. [3] Wikipedia, "Inverse Gaussian distribution" https://en.wikipedia.org/wiki/Inverse_Gaussian_distribution Examples -------- Draw values from the distribution and plot the histogram: >>> import matplotlib.pyplot as plt >>> h = plt.hist(np.random.wald(3, 2, 100000), bins=200, density=True) >>> plt.show() rayleigh(scale=1.0, size=None) Draw samples from a Rayleigh distribution. The :math:`\chi` and Weibull distributions are generalizations of the Rayleigh. .. note:: New code should use the `~numpy.random.Generator.rayleigh` method of a `~numpy.random.Generator` instance instead; please see the :ref:`random-quick-start`. Parameters ---------- scale : float or array_like of floats, optional Scale, also equals the mode. Must be non-negative. Default is 1. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``scale`` is a scalar. Otherwise, ``np.array(scale).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized Rayleigh distribution. See Also -------- random.Generator.rayleigh: which should be used for new code. Notes ----- The probability density function for the Rayleigh distribution is .. math:: P(x;scale) = \frac{x}{scale^2}e^{\frac{-x^2}{2 \cdotp scale^2}} The Rayleigh distribution would arise, for example, if the East and North components of the wind velocity had identical zero-mean Gaussian distributions. Then the wind speed would have a Rayleigh distribution. References ---------- .. [1] Brighton Webs Ltd., "Rayleigh Distribution," https://web.archive.org/web/20090514091424/http://brighton-webs.co.uk:80/distributions/rayleigh.asp .. [2] Wikipedia, "Rayleigh distribution" https://en.wikipedia.org/wiki/Rayleigh_distribution Examples -------- Draw values from the distribution and plot the histogram >>> from matplotlib.pyplot import hist >>> values = hist(np.random.rayleigh(3, 100000), bins=200, density=True) Wave heights tend to follow a Rayleigh distribution. If the mean wave height is 1 meter, what fraction of waves are likely to be larger than 3 meters? >>> meanvalue = 1 >>> modevalue = np.sqrt(2 / np.pi) * meanvalue >>> s = np.random.rayleigh(modevalue, 1000000) The percentage of waves larger than 3 meters is: >>> 100.*sum(s>3)/1000000. 0.087300000000000003 # random lognormal(mean=0.0, sigma=1.0, size=None) Draw samples from a log-normal distribution. Draw samples from a log-normal distribution with specified mean, standard deviation, and array shape. Note that the mean and standard deviation are not the values for the distribution itself, but of the underlying normal distribution it is derived from. .. note:: New code should use the `~numpy.random.Generator.lognormal` method of a `~numpy.random.Generator` instance instead; please see the :ref:`random-quick-start`. Parameters ---------- mean : float or array_like of floats, optional Mean value of the underlying normal distribution. Default is 0. sigma : float or array_like of floats, optional Standard deviation of the underlying normal distribution. Must be non-negative. Default is 1. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``mean`` and ``sigma`` are both scalars. Otherwise, ``np.broadcast(mean, sigma).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized log-normal distribution. See Also -------- scipy.stats.lognorm : probability density function, distribution, cumulative density function, etc. random.Generator.lognormal: which should be used for new code. Notes ----- A variable `x` has a log-normal distribution if `log(x)` is normally distributed. The probability density function for the log-normal distribution is: .. math:: p(x) = \frac{1}{\sigma x \sqrt{2\pi}} e^{(-\frac{(ln(x)-\mu)^2}{2\sigma^2})} where :math:`\mu` is the mean and :math:`\sigma` is the standard deviation of the normally distributed logarithm of the variable. A log-normal distribution results if a random variable is the *product* of a large number of independent, identically-distributed variables in the same way that a normal distribution results if the variable is the *sum* of a large number of independent, identically-distributed variables. References ---------- .. [1] Limpert, E., Stahel, W. A., and Abbt, M., "Log-normal Distributions across the Sciences: Keys and Clues," BioScience, Vol. 51, No. 5, May, 2001. https://stat.ethz.ch/~stahel/lognormal/bioscience.pdf .. [2] Reiss, R.D. and Thomas, M., "Statistical Analysis of Extreme Values," Basel: Birkhauser Verlag, 2001, pp. 31-32. Examples -------- Draw samples from the distribution: >>> mu, sigma = 3., 1. # mean and standard deviation >>> s = np.random.lognormal(mu, sigma, 1000) Display the histogram of the samples, along with the probability density function: >>> import matplotlib.pyplot as plt >>> count, bins, ignored = plt.hist(s, 100, density=True, align='mid') >>> x = np.linspace(min(bins), max(bins), 10000) >>> pdf = (np.exp(-(np.log(x) - mu)**2 / (2 * sigma**2)) ... / (x * sigma * np.sqrt(2 * np.pi))) >>> plt.plot(x, pdf, linewidth=2, color='r') >>> plt.axis('tight') >>> plt.show() Demonstrate that taking the products of random samples from a uniform distribution can be fit well by a log-normal probability density function. >>> # Generate a thousand samples: each is the product of 100 random >>> # values, drawn from a normal distribution. >>> b = [] >>> for i in range(1000): ... a = 10. + np.random.standard_normal(100) ... b.append(np.prod(a)) >>> b = np.array(b) / np.min(b) # scale values to be positive >>> count, bins, ignored = plt.hist(b, 100, density=True, align='mid') >>> sigma = np.std(np.log(b)) >>> mu = np.mean(np.log(b)) >>> x = np.linspace(min(bins), max(bins), 10000) >>> pdf = (np.exp(-(np.log(x) - mu)**2 / (2 * sigma**2)) ... / (x * sigma * np.sqrt(2 * np.pi))) >>> plt.plot(x, pdf, color='r', linewidth=2) >>> plt.show() logistic(loc=0.0, scale=1.0, size=None) Draw samples from a logistic distribution. Samples are drawn from a logistic distribution with specified parameters, loc (location or mean, also median), and scale (>0). .. note:: New code should use the `~numpy.random.Generator.logistic` method of a `~numpy.random.Generator` instance instead; please see the :ref:`random-quick-start`. Parameters ---------- loc : float or array_like of floats, optional Parameter of the distribution. Default is 0. scale : float or array_like of floats, optional Parameter of the distribution. Must be non-negative. Default is 1. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``loc`` and ``scale`` are both scalars. Otherwise, ``np.broadcast(loc, scale).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized logistic distribution. See Also -------- scipy.stats.logistic : probability density function, distribution or cumulative density function, etc. random.Generator.logistic: which should be used for new code. Notes ----- The probability density for the Logistic distribution is .. math:: P(x) = P(x) = \frac{e^{-(x-\mu)/s}}{s(1+e^{-(x-\mu)/s})^2}, where :math:`\mu` = location and :math:`s` = scale. The Logistic distribution is used in Extreme Value problems where it can act as a mixture of Gumbel distributions, in Epidemiology, and by the World Chess Federation (FIDE) where it is used in the Elo ranking system, assuming the performance of each player is a logistically distributed random variable. References ---------- .. [1] Reiss, R.-D. and Thomas M. (2001), "Statistical Analysis of Extreme Values, from Insurance, Finance, Hydrology and Other Fields," Birkhauser Verlag, Basel, pp 132-133. .. [2] Weisstein, Eric W. "Logistic Distribution." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/LogisticDistribution.html .. [3] Wikipedia, "Logistic-distribution", https://en.wikipedia.org/wiki/Logistic_distribution Examples -------- Draw samples from the distribution: >>> loc, scale = 10, 1 >>> s = np.random.logistic(loc, scale, 10000) >>> import matplotlib.pyplot as plt >>> count, bins, ignored = plt.hist(s, bins=50) # plot against distribution >>> def logist(x, loc, scale): ... return np.exp((loc-x)/scale)/(scale*(1+np.exp((loc-x)/scale))**2) >>> lgst_val = logist(bins, loc, scale) >>> plt.plot(bins, lgst_val * count.max() / lgst_val.max()) >>> plt.show() gumbel(loc=0.0, scale=1.0, size=None) Draw samples from a Gumbel distribution. Draw samples from a Gumbel distribution with specified location and scale. For more information on the Gumbel distribution, see Notes and References below. .. note:: New code should use the `~numpy.random.Generator.gumbel` method of a `~numpy.random.Generator` instance instead; please see the :ref:`random-quick-start`. Parameters ---------- loc : float or array_like of floats, optional The location of the mode of the distribution. Default is 0. scale : float or array_like of floats, optional The scale parameter of the distribution. Default is 1. Must be non- negative. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``loc`` and ``scale`` are both scalars. Otherwise, ``np.broadcast(loc, scale).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized Gumbel distribution. See Also -------- scipy.stats.gumbel_l scipy.stats.gumbel_r scipy.stats.genextreme weibull random.Generator.gumbel: which should be used for new code. Notes ----- The Gumbel (or Smallest Extreme Value (SEV) or the Smallest Extreme Value Type I) distribution is one of a class of Generalized Extreme Value (GEV) distributions used in modeling extreme value problems. The Gumbel is a special case of the Extreme Value Type I distribution for maximums from distributions with "exponential-like" tails. The probability density for the Gumbel distribution is .. math:: p(x) = \frac{e^{-(x - \mu)/ \beta}}{\beta} e^{ -e^{-(x - \mu)/ \beta}}, where :math:`\mu` is the mode, a location parameter, and :math:`\beta` is the scale parameter. The Gumbel (named for German mathematician Emil Julius Gumbel) was used very early in the hydrology literature, for modeling the occurrence of flood events. It is also used for modeling maximum wind speed and rainfall rates. It is a "fat-tailed" distribution - the probability of an event in the tail of the distribution is larger than if one used a Gaussian, hence the surprisingly frequent occurrence of 100-year floods. Floods were initially modeled as a Gaussian process, which underestimated the frequency of extreme events. It is one of a class of extreme value distributions, the Generalized Extreme Value (GEV) distributions, which also includes the Weibull and Frechet. The function has a mean of :math:`\mu + 0.57721\beta` and a variance of :math:`\frac{\pi^2}{6}\beta^2`. References ---------- .. [1] Gumbel, E. J., "Statistics of Extremes," New York: Columbia University Press, 1958. .. [2] Reiss, R.-D. and Thomas, M., "Statistical Analysis of Extreme Values from Insurance, Finance, Hydrology and Other Fields," Basel: Birkhauser Verlag, 2001. Examples -------- Draw samples from the distribution: >>> mu, beta = 0, 0.1 # location and scale >>> s = np.random.gumbel(mu, beta, 1000) Display the histogram of the samples, along with the probability density function: >>> import matplotlib.pyplot as plt >>> count, bins, ignored = plt.hist(s, 30, density=True) >>> plt.plot(bins, (1/beta)*np.exp(-(bins - mu)/beta) ... * np.exp( -np.exp( -(bins - mu) /beta) ), ... linewidth=2, color='r') >>> plt.show() Show how an extreme value distribution can arise from a Gaussian process and compare to a Gaussian: >>> means = [] >>> maxima = [] >>> for i in range(0,1000) : ... a = np.random.normal(mu, beta, 1000) ... means.append(a.mean()) ... maxima.append(a.max()) >>> count, bins, ignored = plt.hist(maxima, 30, density=True) >>> beta = np.std(maxima) * np.sqrt(6) / np.pi >>> mu = np.mean(maxima) - 0.57721*beta >>> plt.plot(bins, (1/beta)*np.exp(-(bins - mu)/beta) ... * np.exp(-np.exp(-(bins - mu)/beta)), ... linewidth=2, color='r') >>> plt.plot(bins, 1/(beta * np.sqrt(2 * np.pi)) ... * np.exp(-(bins - mu)**2 / (2 * beta**2)), ... linewidth=2, color='g') >>> plt.show() laplace(loc=0.0, scale=1.0, size=None) Draw samples from the Laplace or double exponential distribution with specified location (or mean) and scale (decay). The Laplace distribution is similar to the Gaussian/normal distribution, but is sharper at the peak and has fatter tails. It represents the difference between two independent, identically distributed exponential random variables. .. note:: New code should use the `~numpy.random.Generator.laplace` method of a `~numpy.random.Generator` instance instead; please see the :ref:`random-quick-start`. Parameters ---------- loc : float or array_like of floats, optional The position, :math:`\mu`, of the distribution peak. Default is 0. scale : float or array_like of floats, optional :math:`\lambda`, the exponential decay. Default is 1. Must be non- negative. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``loc`` and ``scale`` are both scalars. Otherwise, ``np.broadcast(loc, scale).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized Laplace distribution. See Also -------- random.Generator.laplace: which should be used for new code. Notes ----- It has the probability density function .. math:: f(x; \mu, \lambda) = \frac{1}{2\lambda} \exp\left(-\frac{|x - \mu|}{\lambda}\right). The first law of Laplace, from 1774, states that the frequency of an error can be expressed as an exponential function of the absolute magnitude of the error, which leads to the Laplace distribution. For many problems in economics and health sciences, this distribution seems to model the data better than the standard Gaussian distribution. References ---------- .. [1] Abramowitz, M. and Stegun, I. A. (Eds.). "Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing," New York: Dover, 1972. .. [2] Kotz, Samuel, et. al. "The Laplace Distribution and Generalizations, " Birkhauser, 2001. .. [3] Weisstein, Eric W. "Laplace Distribution." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/LaplaceDistribution.html .. [4] Wikipedia, "Laplace distribution", https://en.wikipedia.org/wiki/Laplace_distribution Examples -------- Draw samples from the distribution >>> loc, scale = 0., 1. >>> s = np.random.laplace(loc, scale, 1000) Display the histogram of the samples, along with the probability density function: >>> import matplotlib.pyplot as plt >>> count, bins, ignored = plt.hist(s, 30, density=True) >>> x = np.arange(-8., 8., .01) >>> pdf = np.exp(-abs(x-loc)/scale)/(2.*scale) >>> plt.plot(x, pdf) Plot Gaussian for comparison: >>> g = (1/(scale * np.sqrt(2 * np.pi)) * ... np.exp(-(x - loc)**2 / (2 * scale**2))) >>> plt.plot(x,g) power(a, size=None) Draws samples in [0, 1] from a power distribution with positive exponent a - 1. Also known as the power function distribution. .. note:: New code should use the `~numpy.random.Generator.power` method of a `~numpy.random.Generator` instance instead; please see the :ref:`random-quick-start`. Parameters ---------- a : float or array_like of floats Parameter of the distribution. Must be non-negative. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``a`` is a scalar. Otherwise, ``np.array(a).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized power distribution. Raises ------ ValueError If a <= 0. See Also -------- random.Generator.power: which should be used for new code. Notes ----- The probability density function is .. math:: P(x; a) = ax^{a-1}, 0 \le x \le 1, a>0. The power function distribution is just the inverse of the Pareto distribution. It may also be seen as a special case of the Beta distribution. It is used, for example, in modeling the over-reporting of insurance claims. References ---------- .. [1] Christian Kleiber, Samuel Kotz, "Statistical size distributions in economics and actuarial sciences", Wiley, 2003. .. [2] Heckert, N. A. and Filliben, James J. "NIST Handbook 148: Dataplot Reference Manual, Volume 2: Let Subcommands and Library Functions", National Institute of Standards and Technology Handbook Series, June 2003. https://www.itl.nist.gov/div898/software/dataplot/refman2/auxillar/powpdf.pdf Examples -------- Draw samples from the distribution: >>> a = 5. # shape >>> samples = 1000 >>> s = np.random.power(a, samples) Display the histogram of the samples, along with the probability density function: >>> import matplotlib.pyplot as plt >>> count, bins, ignored = plt.hist(s, bins=30) >>> x = np.linspace(0, 1, 100) >>> y = a*x**(a-1.) >>> normed_y = samples*np.diff(bins)[0]*y >>> plt.plot(x, normed_y) >>> plt.show() Compare the power function distribution to the inverse of the Pareto. >>> from scipy import stats # doctest: +SKIP >>> rvs = np.random.power(5, 1000000) >>> rvsp = np.random.pareto(5, 1000000) >>> xx = np.linspace(0,1,100) >>> powpdf = stats.powerlaw.pdf(xx,5) # doctest: +SKIP >>> plt.figure() >>> plt.hist(rvs, bins=50, density=True) >>> plt.plot(xx,powpdf,'r-') # doctest: +SKIP >>> plt.title('np.random.power(5)') >>> plt.figure() >>> plt.hist(1./(1.+rvsp), bins=50, density=True) >>> plt.plot(xx,powpdf,'r-') # doctest: +SKIP >>> plt.title('inverse of 1 + np.random.pareto(5)') >>> plt.figure() >>> plt.hist(1./(1.+rvsp), bins=50, density=True) >>> plt.plot(xx,powpdf,'r-') # doctest: +SKIP >>> plt.title('inverse of stats.pareto(5)') weibull(a, size=None) Draw samples from a Weibull distribution. Draw samples from a 1-parameter Weibull distribution with the given shape parameter `a`. .. math:: X = (-ln(U))^{1/a} Here, U is drawn from the uniform distribution over (0,1]. The more common 2-parameter Weibull, including a scale parameter :math:`\lambda` is just :math:`X = \lambda(-ln(U))^{1/a}`. .. note:: New code should use the `~numpy.random.Generator.weibull` method of a `~numpy.random.Generator` instance instead; please see the :ref:`random-quick-start`. Parameters ---------- a : float or array_like of floats Shape parameter of the distribution. Must be nonnegative. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``a`` is a scalar. Otherwise, ``np.array(a).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized Weibull distribution. See Also -------- scipy.stats.weibull_max scipy.stats.weibull_min scipy.stats.genextreme gumbel random.Generator.weibull: which should be used for new code. Notes ----- The Weibull (or Type III asymptotic extreme value distribution for smallest values, SEV Type III, or Rosin-Rammler distribution) is one of a class of Generalized Extreme Value (GEV) distributions used in modeling extreme value problems. This class includes the Gumbel and Frechet distributions. The probability density for the Weibull distribution is .. math:: p(x) = \frac{a} {\lambda}(\frac{x}{\lambda})^{a-1}e^{-(x/\lambda)^a}, where :math:`a` is the shape and :math:`\lambda` the scale. The function has its peak (the mode) at :math:`\lambda(\frac{a-1}{a})^{1/a}`. When ``a = 1``, the Weibull distribution reduces to the exponential distribution. References ---------- .. [1] Waloddi Weibull, Royal Technical University, Stockholm, 1939 "A Statistical Theory Of The Strength Of Materials", Ingeniorsvetenskapsakademiens Handlingar Nr 151, 1939, Generalstabens Litografiska Anstalts Forlag, Stockholm. .. [2] Waloddi Weibull, "A Statistical Distribution Function of Wide Applicability", Journal Of Applied Mechanics ASME Paper 1951. .. [3] Wikipedia, "Weibull distribution", https://en.wikipedia.org/wiki/Weibull_distribution Examples -------- Draw samples from the distribution: >>> a = 5. # shape >>> s = np.random.weibull(a, 1000) Display the histogram of the samples, along with the probability density function: >>> import matplotlib.pyplot as plt >>> x = np.arange(1,100.)/50. >>> def weib(x,n,a): ... return (a / n) * (x / n)**(a - 1) * np.exp(-(x / n)**a) >>> count, bins, ignored = plt.hist(np.random.weibull(5.,1000)) >>> x = np.arange(1,100.)/50. >>> scale = count.max()/weib(x, 1., 5.).max() >>> plt.plot(x, weib(x, 1., 5.)*scale) >>> plt.show() pareto(a, size=None) Draw samples from a Pareto II or Lomax distribution with specified shape. The Lomax or Pareto II distribution is a shifted Pareto distribution. The classical Pareto distribution can be obtained from the Lomax distribution by adding 1 and multiplying by the scale parameter ``m`` (see Notes). The smallest value of the Lomax distribution is zero while for the classical Pareto distribution it is ``mu``, where the standard Pareto distribution has location ``mu = 1``. Lomax can also be considered as a simplified version of the Generalized Pareto distribution (available in SciPy), with the scale set to one and the location set to zero. The Pareto distribution must be greater than zero, and is unbounded above. It is also known as the "80-20 rule". In this distribution, 80 percent of the weights are in the lowest 20 percent of the range, while the other 20 percent fill the remaining 80 percent of the range. .. note:: New code should use the `~numpy.random.Generator.pareto` method of a `~numpy.random.Generator` instance instead; please see the :ref:`random-quick-start`. Parameters ---------- a : float or array_like of floats Shape of the distribution. Must be positive. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``a`` is a scalar. Otherwise, ``np.array(a).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized Pareto distribution. See Also -------- scipy.stats.lomax : probability density function, distribution or cumulative density function, etc. scipy.stats.genpareto : probability density function, distribution or cumulative density function, etc. random.Generator.pareto: which should be used for new code. Notes ----- The probability density for the Pareto distribution is .. math:: p(x) = \frac{am^a}{x^{a+1}} where :math:`a` is the shape and :math:`m` the scale. The Pareto distribution, named after the Italian economist Vilfredo Pareto, is a power law probability distribution useful in many real world problems. Outside the field of economics it is generally referred to as the Bradford distribution. Pareto developed the distribution to describe the distribution of wealth in an economy. It has also found use in insurance, web page access statistics, oil field sizes, and many other problems, including the download frequency for projects in Sourceforge [1]_. It is one of the so-called "fat-tailed" distributions. References ---------- .. [1] Francis Hunt and Paul Johnson, On the Pareto Distribution of Sourceforge projects. .. [2] Pareto, V. (1896). Course of Political Economy. Lausanne. .. [3] Reiss, R.D., Thomas, M.(2001), Statistical Analysis of Extreme Values, Birkhauser Verlag, Basel, pp 23-30. .. [4] Wikipedia, "Pareto distribution", https://en.wikipedia.org/wiki/Pareto_distribution Examples -------- Draw samples from the distribution: >>> a, m = 3., 2. # shape and mode >>> s = (np.random.pareto(a, 1000) + 1) * m Display the histogram of the samples, along with the probability density function: >>> import matplotlib.pyplot as plt >>> count, bins, _ = plt.hist(s, 100, density=True) >>> fit = a*m**a / bins**(a+1) >>> plt.plot(bins, max(count)*fit/max(fit), linewidth=2, color='r') >>> plt.show() vonmises(mu, kappa, size=None) Draw samples from a von Mises distribution. Samples are drawn from a von Mises distribution with specified mode (mu) and dispersion (kappa), on the interval [-pi, pi]. The von Mises distribution (also known as the circular normal distribution) is a continuous probability distribution on the unit circle. It may be thought of as the circular analogue of the normal distribution. .. note:: New code should use the `~numpy.random.Generator.vonmises` method of a `~numpy.random.Generator` instance instead; please see the :ref:`random-quick-start`. Parameters ---------- mu : float or array_like of floats Mode ("center") of the distribution. kappa : float or array_like of floats Dispersion of the distribution, has to be >=0. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``mu`` and ``kappa`` are both scalars. Otherwise, ``np.broadcast(mu, kappa).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized von Mises distribution. See Also -------- scipy.stats.vonmises : probability density function, distribution, or cumulative density function, etc. random.Generator.vonmises: which should be used for new code. Notes ----- The probability density for the von Mises distribution is .. math:: p(x) = \frac{e^{\kappa cos(x-\mu)}}{2\pi I_0(\kappa)}, where :math:`\mu` is the mode and :math:`\kappa` the dispersion, and :math:`I_0(\kappa)` is the modified Bessel function of order 0. The von Mises is named for Richard Edler von Mises, who was born in Austria-Hungary, in what is now the Ukraine. He fled to the United States in 1939 and became a professor at Harvard. He worked in probability theory, aerodynamics, fluid mechanics, and philosophy of science. References ---------- .. [1] Abramowitz, M. and Stegun, I. A. (Eds.). "Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing," New York: Dover, 1972. .. [2] von Mises, R., "Mathematical Theory of Probability and Statistics", New York: Academic Press, 1964. Examples -------- Draw samples from the distribution: >>> mu, kappa = 0.0, 4.0 # mean and dispersion >>> s = np.random.vonmises(mu, kappa, 1000) Display the histogram of the samples, along with the probability density function: >>> import matplotlib.pyplot as plt >>> from scipy.special import i0 # doctest: +SKIP >>> plt.hist(s, 50, density=True) >>> x = np.linspace(-np.pi, np.pi, num=51) >>> y = np.exp(kappa*np.cos(x-mu))/(2*np.pi*i0(kappa)) # doctest: +SKIP >>> plt.plot(x, y, linewidth=2, color='r') # doctest: +SKIP >>> plt.show() standard_t(df, size=None) Draw samples from a standard Student's t distribution with `df` degrees of freedom. A special case of the hyperbolic distribution. As `df` gets large, the result resembles that of the standard normal distribution (`standard_normal`). .. note:: New code should use the `~numpy.random.Generator.standard_t` method of a `~numpy.random.Generator` instance instead; please see the :ref:`random-quick-start`. Parameters ---------- df : float or array_like of floats Degrees of freedom, must be > 0. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``df`` is a scalar. Otherwise, ``np.array(df).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized standard Student's t distribution. See Also -------- random.Generator.standard_t: which should be used for new code. Notes ----- The probability density function for the t distribution is .. math:: P(x, df) = \frac{\Gamma(\frac{df+1}{2})}{\sqrt{\pi df} \Gamma(\frac{df}{2})}\Bigl( 1+\frac{x^2}{df} \Bigr)^{-(df+1)/2} The t test is based on an assumption that the data come from a Normal distribution. The t test provides a way to test whether the sample mean (that is the mean calculated from the data) is a good estimate of the true mean. The derivation of the t-distribution was first published in 1908 by William Gosset while working for the Guinness Brewery in Dublin. Due to proprietary issues, he had to publish under a pseudonym, and so he used the name Student. References ---------- .. [1] Dalgaard, Peter, "Introductory Statistics With R", Springer, 2002. .. [2] Wikipedia, "Student's t-distribution" https://en.wikipedia.org/wiki/Student's_t-distribution Examples -------- From Dalgaard page 83 [1]_, suppose the daily energy intake for 11 women in kilojoules (kJ) is: >>> intake = np.array([5260., 5470, 5640, 6180, 6390, 6515, 6805, 7515, \ ... 7515, 8230, 8770]) Does their energy intake deviate systematically from the recommended value of 7725 kJ? Our null hypothesis will be the absence of deviation, and the alternate hypothesis will be the presence of an effect that could be either positive or negative, hence making our test 2-tailed. Because we are estimating the mean and we have N=11 values in our sample, we have N-1=10 degrees of freedom. We set our significance level to 95% and compute the t statistic using the empirical mean and empirical standard deviation of our intake. We use a ddof of 1 to base the computation of our empirical standard deviation on an unbiased estimate of the variance (note: the final estimate is not unbiased due to the concave nature of the square root). >>> np.mean(intake) 6753.636363636364 >>> intake.std(ddof=1) 1142.1232221373727 >>> t = (np.mean(intake)-7725)/(intake.std(ddof=1)/np.sqrt(len(intake))) >>> t -2.8207540608310198 We draw 1000000 samples from Student's t distribution with the adequate degrees of freedom. >>> import matplotlib.pyplot as plt >>> s = np.random.standard_t(10, size=1000000) >>> h = plt.hist(s, bins=100, density=True) Does our t statistic land in one of the two critical regions found at both tails of the distribution? >>> np.sum(np.abs(t) < np.abs(s)) / float(len(s)) 0.018318 #random < 0.05, statistic is in critical region The probability value for this 2-tailed test is about 1.83%, which is lower than the 5% pre-determined significance threshold. Therefore, the probability of observing values as extreme as our intake conditionally on the null hypothesis being true is too low, and we reject the null hypothesis of no deviation. standard_cauchy(size=None) Draw samples from a standard Cauchy distribution with mode = 0. Also known as the Lorentz distribution. .. note:: New code should use the `~numpy.random.Generator.standard_cauchy` method of a `~numpy.random.Generator` instance instead; please see the :ref:`random-quick-start`. Parameters ---------- size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. Default is None, in which case a single value is returned. Returns ------- samples : ndarray or scalar The drawn samples. See Also -------- random.Generator.standard_cauchy: which should be used for new code. Notes ----- The probability density function for the full Cauchy distribution is .. math:: P(x; x_0, \gamma) = \frac{1}{\pi \gamma \bigl[ 1+ (\frac{x-x_0}{\gamma})^2 \bigr] } and the Standard Cauchy distribution just sets :math:`x_0=0` and :math:`\gamma=1` The Cauchy distribution arises in the solution to the driven harmonic oscillator problem, and also describes spectral line broadening. It also describes the distribution of values at which a line tilted at a random angle will cut the x axis. When studying hypothesis tests that assume normality, seeing how the tests perform on data from a Cauchy distribution is a good indicator of their sensitivity to a heavy-tailed distribution, since the Cauchy looks very much like a Gaussian distribution, but with heavier tails. References ---------- .. [1] NIST/SEMATECH e-Handbook of Statistical Methods, "Cauchy Distribution", https://www.itl.nist.gov/div898/handbook/eda/section3/eda3663.htm .. [2] Weisstein, Eric W. "Cauchy Distribution." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/CauchyDistribution.html .. [3] Wikipedia, "Cauchy distribution" https://en.wikipedia.org/wiki/Cauchy_distribution Examples -------- Draw samples and plot the distribution: >>> import matplotlib.pyplot as plt >>> s = np.random.standard_cauchy(1000000) >>> s = s[(s>-25) & (s<25)] # truncate distribution so it plots well >>> plt.hist(s, bins=100) >>> plt.show() noncentral_chisquare(df, nonc, size=None) Draw samples from a noncentral chi-square distribution. The noncentral :math:`\chi^2` distribution is a generalization of the :math:`\chi^2` distribution. .. note:: New code should use the `~numpy.random.Generator.noncentral_chisquare` method of a `~numpy.random.Generator` instance instead; please see the :ref:`random-quick-start`. Parameters ---------- df : float or array_like of floats Degrees of freedom, must be > 0. .. versionchanged:: 1.10.0 Earlier NumPy versions required dfnum > 1. nonc : float or array_like of floats Non-centrality, must be non-negative. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``df`` and ``nonc`` are both scalars. Otherwise, ``np.broadcast(df, nonc).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized noncentral chi-square distribution. See Also -------- random.Generator.noncentral_chisquare: which should be used for new code. Notes ----- The probability density function for the noncentral Chi-square distribution is .. math:: P(x;df,nonc) = \sum^{\infty}_{i=0} \frac{e^{-nonc/2}(nonc/2)^{i}}{i!} P_{Y_{df+2i}}(x), where :math:`Y_{q}` is the Chi-square with q degrees of freedom. References ---------- .. [1] Wikipedia, "Noncentral chi-squared distribution" https://en.wikipedia.org/wiki/Noncentral_chi-squared_distribution Examples -------- Draw values from the distribution and plot the histogram >>> import matplotlib.pyplot as plt >>> values = plt.hist(np.random.noncentral_chisquare(3, 20, 100000), ... bins=200, density=True) >>> plt.show() Draw values from a noncentral chisquare with very small noncentrality, and compare to a chisquare. >>> plt.figure() >>> values = plt.hist(np.random.noncentral_chisquare(3, .0000001, 100000), ... bins=np.arange(0., 25, .1), density=True) >>> values2 = plt.hist(np.random.chisquare(3, 100000), ... bins=np.arange(0., 25, .1), density=True) >>> plt.plot(values[1][0:-1], values[0]-values2[0], 'ob') >>> plt.show() Demonstrate how large values of non-centrality lead to a more symmetric distribution. >>> plt.figure() >>> values = plt.hist(np.random.noncentral_chisquare(3, 20, 100000), ... bins=200, density=True) >>> plt.show() chisquare(df, size=None) Draw samples from a chi-square distribution. When `df` independent random variables, each with standard normal distributions (mean 0, variance 1), are squared and summed, the resulting distribution is chi-square (see Notes). This distribution is often used in hypothesis testing. .. note:: New code should use the `~numpy.random.Generator.chisquare` method of a `~numpy.random.Generator` instance instead; please see the :ref:`random-quick-start`. Parameters ---------- df : float or array_like of floats Number of degrees of freedom, must be > 0. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``df`` is a scalar. Otherwise, ``np.array(df).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized chi-square distribution. Raises ------ ValueError When `df` <= 0 or when an inappropriate `size` (e.g. ``size=-1``) is given. See Also -------- random.Generator.chisquare: which should be used for new code. Notes ----- The variable obtained by summing the squares of `df` independent, standard normally distributed random variables: .. math:: Q = \sum_{i=0}^{\mathtt{df}} X^2_i is chi-square distributed, denoted .. math:: Q \sim \chi^2_k. The probability density function of the chi-squared distribution is .. math:: p(x) = \frac{(1/2)^{k/2}}{\Gamma(k/2)} x^{k/2 - 1} e^{-x/2}, where :math:`\Gamma` is the gamma function, .. math:: \Gamma(x) = \int_0^{-\infty} t^{x - 1} e^{-t} dt. References ---------- .. [1] NIST "Engineering Statistics Handbook" https://www.itl.nist.gov/div898/handbook/eda/section3/eda3666.htm Examples -------- >>> np.random.chisquare(2,4) array([ 1.89920014, 9.00867716, 3.13710533, 5.62318272]) # random noncentral_f(dfnum, dfden, nonc, size=None) Draw samples from the noncentral F distribution. Samples are drawn from an F distribution with specified parameters, `dfnum` (degrees of freedom in numerator) and `dfden` (degrees of freedom in denominator), where both parameters > 1. `nonc` is the non-centrality parameter. .. note:: New code should use the `~numpy.random.Generator.noncentral_f` method of a `~numpy.random.Generator` instance instead; please see the :ref:`random-quick-start`. Parameters ---------- dfnum : float or array_like of floats Numerator degrees of freedom, must be > 0. .. versionchanged:: 1.14.0 Earlier NumPy versions required dfnum > 1. dfden : float or array_like of floats Denominator degrees of freedom, must be > 0. nonc : float or array_like of floats Non-centrality parameter, the sum of the squares of the numerator means, must be >= 0. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``dfnum``, ``dfden``, and ``nonc`` are all scalars. Otherwise, ``np.broadcast(dfnum, dfden, nonc).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized noncentral Fisher distribution. See Also -------- random.Generator.noncentral_f: which should be used for new code. Notes ----- When calculating the power of an experiment (power = probability of rejecting the null hypothesis when a specific alternative is true) the non-central F statistic becomes important. When the null hypothesis is true, the F statistic follows a central F distribution. When the null hypothesis is not true, then it follows a non-central F statistic. References ---------- .. [1] Weisstein, Eric W. "Noncentral F-Distribution." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/NoncentralF-Distribution.html .. [2] Wikipedia, "Noncentral F-distribution", https://en.wikipedia.org/wiki/Noncentral_F-distribution Examples -------- In a study, testing for a specific alternative to the null hypothesis requires use of the Noncentral F distribution. We need to calculate the area in the tail of the distribution that exceeds the value of the F distribution for the null hypothesis. We'll plot the two probability distributions for comparison. >>> dfnum = 3 # between group deg of freedom >>> dfden = 20 # within groups degrees of freedom >>> nonc = 3.0 >>> nc_vals = np.random.noncentral_f(dfnum, dfden, nonc, 1000000) >>> NF = np.histogram(nc_vals, bins=50, density=True) >>> c_vals = np.random.f(dfnum, dfden, 1000000) >>> F = np.histogram(c_vals, bins=50, density=True) >>> import matplotlib.pyplot as plt >>> plt.plot(F[1][1:], F[0]) >>> plt.plot(NF[1][1:], NF[0]) >>> plt.show() f(dfnum, dfden, size=None) Draw samples from an F distribution. Samples are drawn from an F distribution with specified parameters, `dfnum` (degrees of freedom in numerator) and `dfden` (degrees of freedom in denominator), where both parameters must be greater than zero. The random variate of the F distribution (also known as the Fisher distribution) is a continuous probability distribution that arises in ANOVA tests, and is the ratio of two chi-square variates. .. note:: New code should use the `~numpy.random.Generator.f` method of a `~numpy.random.Generator` instance instead; please see the :ref:`random-quick-start`. Parameters ---------- dfnum : float or array_like of floats Degrees of freedom in numerator, must be > 0. dfden : float or array_like of float Degrees of freedom in denominator, must be > 0. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``dfnum`` and ``dfden`` are both scalars. Otherwise, ``np.broadcast(dfnum, dfden).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized Fisher distribution. See Also -------- scipy.stats.f : probability density function, distribution or cumulative density function, etc. random.Generator.f: which should be used for new code. Notes ----- The F statistic is used to compare in-group variances to between-group variances. Calculating the distribution depends on the sampling, and so it is a function of the respective degrees of freedom in the problem. The variable `dfnum` is the number of samples minus one, the between-groups degrees of freedom, while `dfden` is the within-groups degrees of freedom, the sum of the number of samples in each group minus the number of groups. References ---------- .. [1] Glantz, Stanton A. "Primer of Biostatistics.", McGraw-Hill, Fifth Edition, 2002. .. [2] Wikipedia, "F-distribution", https://en.wikipedia.org/wiki/F-distribution Examples -------- An example from Glantz[1], pp 47-40: Two groups, children of diabetics (25 people) and children from people without diabetes (25 controls). Fasting blood glucose was measured, case group had a mean value of 86.1, controls had a mean value of 82.2. Standard deviations were 2.09 and 2.49 respectively. Are these data consistent with the null hypothesis that the parents diabetic status does not affect their children's blood glucose levels? Calculating the F statistic from the data gives a value of 36.01. Draw samples from the distribution: >>> dfnum = 1. # between group degrees of freedom >>> dfden = 48. # within groups degrees of freedom >>> s = np.random.f(dfnum, dfden, 1000) The lower bound for the top 1% of the samples is : >>> np.sort(s)[-10] 7.61988120985 # random So there is about a 1% chance that the F statistic will exceed 7.62, the measured value is 36, so the null hypothesis is rejected at the 1% level. gamma(shape, scale=1.0, size=None) Draw samples from a Gamma distribution. Samples are drawn from a Gamma distribution with specified parameters, `shape` (sometimes designated "k") and `scale` (sometimes designated "theta"), where both parameters are > 0. .. note:: New code should use the `~numpy.random.Generator.gamma` method of a `~numpy.random.Generator` instance instead; please see the :ref:`random-quick-start`. Parameters ---------- shape : float or array_like of floats The shape of the gamma distribution. Must be non-negative. scale : float or array_like of floats, optional The scale of the gamma distribution. Must be non-negative. Default is equal to 1. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``shape`` and ``scale`` are both scalars. Otherwise, ``np.broadcast(shape, scale).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized gamma distribution. See Also -------- scipy.stats.gamma : probability density function, distribution or cumulative density function, etc. random.Generator.gamma: which should be used for new code. Notes ----- The probability density for the Gamma distribution is .. math:: p(x) = x^{k-1}\frac{e^{-x/\theta}}{\theta^k\Gamma(k)}, where :math:`k` is the shape and :math:`\theta` the scale, and :math:`\Gamma` is the Gamma function. The Gamma distribution is often used to model the times to failure of electronic components, and arises naturally in processes for which the waiting times between Poisson distributed events are relevant. References ---------- .. [1] Weisstein, Eric W. "Gamma Distribution." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/GammaDistribution.html .. [2] Wikipedia, "Gamma distribution", https://en.wikipedia.org/wiki/Gamma_distribution Examples -------- Draw samples from the distribution: >>> shape, scale = 2., 2. # mean=4, std=2*sqrt(2) >>> s = np.random.gamma(shape, scale, 1000) Display the histogram of the samples, along with the probability density function: >>> import matplotlib.pyplot as plt >>> import scipy.special as sps # doctest: +SKIP >>> count, bins, ignored = plt.hist(s, 50, density=True) >>> y = bins**(shape-1)*(np.exp(-bins/scale) / # doctest: +SKIP ... (sps.gamma(shape)*scale**shape)) >>> plt.plot(bins, y, linewidth=2, color='r') # doctest: +SKIP >>> plt.show() standard_gamma(shape, size=None) Draw samples from a standard Gamma distribution. Samples are drawn from a Gamma distribution with specified parameters, shape (sometimes designated "k") and scale=1. .. note:: New code should use the `~numpy.random.Generator.standard_gamma` method of a `~numpy.random.Generator` instance instead; please see the :ref:`random-quick-start`. Parameters ---------- shape : float or array_like of floats Parameter, must be non-negative. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``shape`` is a scalar. Otherwise, ``np.array(shape).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized standard gamma distribution. See Also -------- scipy.stats.gamma : probability density function, distribution or cumulative density function, etc. random.Generator.standard_gamma: which should be used for new code. Notes ----- The probability density for the Gamma distribution is .. math:: p(x) = x^{k-1}\frac{e^{-x/\theta}}{\theta^k\Gamma(k)}, where :math:`k` is the shape and :math:`\theta` the scale, and :math:`\Gamma` is the Gamma function. The Gamma distribution is often used to model the times to failure of electronic components, and arises naturally in processes for which the waiting times between Poisson distributed events are relevant. References ---------- .. [1] Weisstein, Eric W. "Gamma Distribution." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/GammaDistribution.html .. [2] Wikipedia, "Gamma distribution", https://en.wikipedia.org/wiki/Gamma_distribution Examples -------- Draw samples from the distribution: >>> shape, scale = 2., 1. # mean and width >>> s = np.random.standard_gamma(shape, 1000000) Display the histogram of the samples, along with the probability density function: >>> import matplotlib.pyplot as plt >>> import scipy.special as sps # doctest: +SKIP >>> count, bins, ignored = plt.hist(s, 50, density=True) >>> y = bins**(shape-1) * ((np.exp(-bins/scale))/ # doctest: +SKIP ... (sps.gamma(shape) * scale**shape)) >>> plt.plot(bins, y, linewidth=2, color='r') # doctest: +SKIP >>> plt.show() normal(loc=0.0, scale=1.0, size=None) Draw random samples from a normal (Gaussian) distribution. The probability density function of the normal distribution, first derived by De Moivre and 200 years later by both Gauss and Laplace independently [2]_, is often called the bell curve because of its characteristic shape (see the example below). The normal distributions occurs often in nature. For example, it describes the commonly occurring distribution of samples influenced by a large number of tiny, random disturbances, each with its own unique distribution [2]_. .. note:: New code should use the `~numpy.random.Generator.normal` method of a `~numpy.random.Generator` instance instead; please see the :ref:`random-quick-start`. Parameters ---------- loc : float or array_like of floats Mean ("centre") of the distribution. scale : float or array_like of floats Standard deviation (spread or "width") of the distribution. Must be non-negative. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``loc`` and ``scale`` are both scalars. Otherwise, ``np.broadcast(loc, scale).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized normal distribution. See Also -------- scipy.stats.norm : probability density function, distribution or cumulative density function, etc. random.Generator.normal: which should be used for new code. Notes ----- The probability density for the Gaussian distribution is .. math:: p(x) = \frac{1}{\sqrt{ 2 \pi \sigma^2 }} e^{ - \frac{ (x - \mu)^2 } {2 \sigma^2} }, where :math:`\mu` is the mean and :math:`\sigma` the standard deviation. The square of the standard deviation, :math:`\sigma^2`, is called the variance. The function has its peak at the mean, and its "spread" increases with the standard deviation (the function reaches 0.607 times its maximum at :math:`x + \sigma` and :math:`x - \sigma` [2]_). This implies that normal is more likely to return samples lying close to the mean, rather than those far away. References ---------- .. [1] Wikipedia, "Normal distribution", https://en.wikipedia.org/wiki/Normal_distribution .. [2] P. R. Peebles Jr., "Central Limit Theorem" in "Probability, Random Variables and Random Signal Principles", 4th ed., 2001, pp. 51, 51, 125. Examples -------- Draw samples from the distribution: >>> mu, sigma = 0, 0.1 # mean and standard deviation >>> s = np.random.normal(mu, sigma, 1000) Verify the mean and the variance: >>> abs(mu - np.mean(s)) 0.0 # may vary >>> abs(sigma - np.std(s, ddof=1)) 0.1 # may vary Display the histogram of the samples, along with the probability density function: >>> import matplotlib.pyplot as plt >>> count, bins, ignored = plt.hist(s, 30, density=True) >>> plt.plot(bins, 1/(sigma * np.sqrt(2 * np.pi)) * ... np.exp( - (bins - mu)**2 / (2 * sigma**2) ), ... linewidth=2, color='r') >>> plt.show() Two-by-four array of samples from the normal distribution with mean 3 and standard deviation 2.5: >>> np.random.normal(3, 2.5, size=(2, 4)) array([[-4.49401501, 4.00950034, -1.81814867, 7.29718677], # random [ 0.39924804, 4.68456316, 4.99394529, 4.84057254]]) # random standard_normal(size=None) Draw samples from a standard Normal distribution (mean=0, stdev=1). .. note:: New code should use the `~numpy.random.Generator.standard_normal` method of a `~numpy.random.Generator` instance instead; please see the :ref:`random-quick-start`. Parameters ---------- size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. Default is None, in which case a single value is returned. Returns ------- out : float or ndarray A floating-point array of shape ``size`` of drawn samples, or a single sample if ``size`` was not specified. See Also -------- normal : Equivalent function with additional ``loc`` and ``scale`` arguments for setting the mean and standard deviation. random.Generator.standard_normal: which should be used for new code. Notes ----- For random samples from the normal distribution with mean ``mu`` and standard deviation ``sigma``, use one of:: mu + sigma * np.random.standard_normal(size=...) np.random.normal(mu, sigma, size=...) Examples -------- >>> np.random.standard_normal() 2.1923875335537315 #random >>> s = np.random.standard_normal(8000) >>> s array([ 0.6888893 , 0.78096262, -0.89086505, ..., 0.49876311, # random -0.38672696, -0.4685006 ]) # random >>> s.shape (8000,) >>> s = np.random.standard_normal(size=(3, 4, 2)) >>> s.shape (3, 4, 2) Two-by-four array of samples from the normal distribution with mean 3 and standard deviation 2.5: >>> 3 + 2.5 * np.random.standard_normal(size=(2, 4)) array([[-4.49401501, 4.00950034, -1.81814867, 7.29718677], # random [ 0.39924804, 4.68456316, 4.99394529, 4.84057254]]) # random random_integers(low, high=None, size=None) Random integers of type `np.int_` between `low` and `high`, inclusive. Return random integers of type `np.int_` from the "discrete uniform" distribution in the closed interval [`low`, `high`]. If `high` is None (the default), then results are from [1, `low`]. The `np.int_` type translates to the C long integer type and its precision is platform dependent. This function has been deprecated. Use randint instead. .. deprecated:: 1.11.0 Parameters ---------- low : int Lowest (signed) integer to be drawn from the distribution (unless ``high=None``, in which case this parameter is the *highest* such integer). high : int, optional If provided, the largest (signed) integer to be drawn from the distribution (see above for behavior if ``high=None``). size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. Default is None, in which case a single value is returned. Returns ------- out : int or ndarray of ints `size`-shaped array of random integers from the appropriate distribution, or a single such random int if `size` not provided. See Also -------- randint : Similar to `random_integers`, only for the half-open interval [`low`, `high`), and 0 is the lowest value if `high` is omitted. Notes ----- To sample from N evenly spaced floating-point numbers between a and b, use:: a + (b - a) * (np.random.random_integers(N) - 1) / (N - 1.) Examples -------- >>> np.random.random_integers(5) 4 # random >>> type(np.random.random_integers(5)) >>> np.random.random_integers(5, size=(3,2)) array([[5, 4], # random [3, 3], [4, 5]]) Choose five random numbers from the set of five evenly-spaced numbers between 0 and 2.5, inclusive (*i.e.*, from the set :math:`{0, 5/8, 10/8, 15/8, 20/8}`): >>> 2.5 * (np.random.random_integers(5, size=(5,)) - 1) / 4. array([ 0.625, 1.25 , 0.625, 0.625, 2.5 ]) # random Roll two six sided dice 1000 times and sum the results: >>> d1 = np.random.random_integers(1, 6, 1000) >>> d2 = np.random.random_integers(1, 6, 1000) >>> dsums = d1 + d2 Display results as a histogram: >>> import matplotlib.pyplot as plt >>> count, bins, ignored = plt.hist(dsums, 11, density=True) >>> plt.show() randn(d0, d1, ..., dn) Return a sample (or samples) from the "standard normal" distribution. .. note:: This is a convenience function for users porting code from Matlab, and wraps `standard_normal`. That function takes a tuple to specify the size of the output, which is consistent with other NumPy functions like `numpy.zeros` and `numpy.ones`. .. note:: New code should use the `~numpy.random.Generator.standard_normal` method of a `~numpy.random.Generator` instance instead; please see the :ref:`random-quick-start`. If positive int_like arguments are provided, `randn` generates an array of shape ``(d0, d1, ..., dn)``, filled with random floats sampled from a univariate "normal" (Gaussian) distribution of mean 0 and variance 1. A single float randomly sampled from the distribution is returned if no argument is provided. Parameters ---------- d0, d1, ..., dn : int, optional The dimensions of the returned array, must be non-negative. If no argument is given a single Python float is returned. Returns ------- Z : ndarray or float A ``(d0, d1, ..., dn)``-shaped array of floating-point samples from the standard normal distribution, or a single such float if no parameters were supplied. See Also -------- standard_normal : Similar, but takes a tuple as its argument. normal : Also accepts mu and sigma arguments. random.Generator.standard_normal: which should be used for new code. Notes ----- For random samples from the normal distribution with mean ``mu`` and standard deviation ``sigma``, use:: sigma * np.random.randn(...) + mu Examples -------- >>> np.random.randn() 2.1923875335537315 # random Two-by-four array of samples from the normal distribution with mean 3 and standard deviation 2.5: >>> 3 + 2.5 * np.random.randn(2, 4) array([[-4.49401501, 4.00950034, -1.81814867, 7.29718677], # random [ 0.39924804, 4.68456316, 4.99394529, 4.84057254]]) # random rand(d0, d1, ..., dn) Random values in a given shape. .. note:: This is a convenience function for users porting code from Matlab, and wraps `random_sample`. That function takes a tuple to specify the size of the output, which is consistent with other NumPy functions like `numpy.zeros` and `numpy.ones`. Create an array of the given shape and populate it with random samples from a uniform distribution over ``[0, 1)``. Parameters ---------- d0, d1, ..., dn : int, optional The dimensions of the returned array, must be non-negative. If no argument is given a single Python float is returned. Returns ------- out : ndarray, shape ``(d0, d1, ..., dn)`` Random values. See Also -------- random Examples -------- >>> np.random.rand(3,2) array([[ 0.14022471, 0.96360618], #random [ 0.37601032, 0.25528411], #random [ 0.49313049, 0.94909878]]) #random uniform(low=0.0, high=1.0, size=None) Draw samples from a uniform distribution. Samples are uniformly distributed over the half-open interval ``[low, high)`` (includes low, but excludes high). In other words, any value within the given interval is equally likely to be drawn by `uniform`. .. note:: New code should use the `~numpy.random.Generator.uniform` method of a `~numpy.random.Generator` instance instead; please see the :ref:`random-quick-start`. Parameters ---------- low : float or array_like of floats, optional Lower boundary of the output interval. All values generated will be greater than or equal to low. The default value is 0. high : float or array_like of floats Upper boundary of the output interval. All values generated will be less than or equal to high. The high limit may be included in the returned array of floats due to floating-point rounding in the equation ``low + (high-low) * random_sample()``. The default value is 1.0. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``low`` and ``high`` are both scalars. Otherwise, ``np.broadcast(low, high).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized uniform distribution. See Also -------- randint : Discrete uniform distribution, yielding integers. random_integers : Discrete uniform distribution over the closed interval ``[low, high]``. random_sample : Floats uniformly distributed over ``[0, 1)``. random : Alias for `random_sample`. rand : Convenience function that accepts dimensions as input, e.g., ``rand(2,2)`` would generate a 2-by-2 array of floats, uniformly distributed over ``[0, 1)``. random.Generator.uniform: which should be used for new code. Notes ----- The probability density function of the uniform distribution is .. math:: p(x) = \frac{1}{b - a} anywhere within the interval ``[a, b)``, and zero elsewhere. When ``high`` == ``low``, values of ``low`` will be returned. If ``high`` < ``low``, the results are officially undefined and may eventually raise an error, i.e. do not rely on this function to behave when passed arguments satisfying that inequality condition. The ``high`` limit may be included in the returned array of floats due to floating-point rounding in the equation ``low + (high-low) * random_sample()``. For example: >>> x = np.float32(5*0.99999999) >>> x 5.0 Examples -------- Draw samples from the distribution: >>> s = np.random.uniform(-1,0,1000) All values are within the given interval: >>> np.all(s >= -1) True >>> np.all(s < 0) True Display the histogram of the samples, along with the probability density function: >>> import matplotlib.pyplot as plt >>> count, bins, ignored = plt.hist(s, 15, density=True) >>> plt.plot(bins, np.ones_like(bins), linewidth=2, color='r') >>> plt.show() choice(a, size=None, replace=True, p=None) Generates a random sample from a given 1-D array .. versionadded:: 1.7.0 .. note:: New code should use the `~numpy.random.Generator.choice` method of a `~numpy.random.Generator` instance instead; please see the :ref:`random-quick-start`. Parameters ---------- a : 1-D array-like or int If an ndarray, a random sample is generated from its elements. If an int, the random sample is generated as if it were ``np.arange(a)`` size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. Default is None, in which case a single value is returned. replace : boolean, optional Whether the sample is with or without replacement. Default is True, meaning that a value of ``a`` can be selected multiple times. p : 1-D array-like, optional The probabilities associated with each entry in a. If not given, the sample assumes a uniform distribution over all entries in ``a``. Returns ------- samples : single item or ndarray The generated random samples Raises ------ ValueError If a is an int and less than zero, if a or p are not 1-dimensional, if a is an array-like of size 0, if p is not a vector of probabilities, if a and p have different lengths, or if replace=False and the sample size is greater than the population size See Also -------- randint, shuffle, permutation random.Generator.choice: which should be used in new code Notes ----- Setting user-specified probabilities through ``p`` uses a more general but less efficient sampler than the default. The general sampler produces a different sample than the optimized sampler even if each element of ``p`` is 1 / len(a). Sampling random rows from a 2-D array is not possible with this function, but is possible with `Generator.choice` through its ``axis`` keyword. Examples -------- Generate a uniform random sample from np.arange(5) of size 3: >>> np.random.choice(5, 3) array([0, 3, 4]) # random >>> #This is equivalent to np.random.randint(0,5,3) Generate a non-uniform random sample from np.arange(5) of size 3: >>> np.random.choice(5, 3, p=[0.1, 0, 0.3, 0.6, 0]) array([3, 3, 0]) # random Generate a uniform random sample from np.arange(5) of size 3 without replacement: >>> np.random.choice(5, 3, replace=False) array([3,1,0]) # random >>> #This is equivalent to np.random.permutation(np.arange(5))[:3] Generate a non-uniform random sample from np.arange(5) of size 3 without replacement: >>> np.random.choice(5, 3, replace=False, p=[0.1, 0, 0.3, 0.6, 0]) array([2, 3, 0]) # random Any of the above can be repeated with an arbitrary array-like instead of just integers. For instance: >>> aa_milne_arr = ['pooh', 'rabbit', 'piglet', 'Christopher'] >>> np.random.choice(aa_milne_arr, 5, p=[0.5, 0.1, 0.1, 0.3]) array(['pooh', 'pooh', 'pooh', 'Christopher', 'piglet'], # random dtype='>> np.random.bytes(10) b' eh\x85\x022SZ\xbf\xa4' #random randint(low, high=None, size=None, dtype=int) Return random integers from `low` (inclusive) to `high` (exclusive). Return random integers from the "discrete uniform" distribution of the specified dtype in the "half-open" interval [`low`, `high`). If `high` is None (the default), then results are from [0, `low`). .. note:: New code should use the `~numpy.random.Generator.integers` method of a `~numpy.random.Generator` instance instead; please see the :ref:`random-quick-start`. Parameters ---------- low : int or array-like of ints Lowest (signed) integers to be drawn from the distribution (unless ``high=None``, in which case this parameter is one above the *highest* such integer). high : int or array-like of ints, optional If provided, one above the largest (signed) integer to be drawn from the distribution (see above for behavior if ``high=None``). If array-like, must contain integer values size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. Default is None, in which case a single value is returned. dtype : dtype, optional Desired dtype of the result. Byteorder must be native. The default value is int. .. versionadded:: 1.11.0 Returns ------- out : int or ndarray of ints `size`-shaped array of random integers from the appropriate distribution, or a single such random int if `size` not provided. See Also -------- random_integers : similar to `randint`, only for the closed interval [`low`, `high`], and 1 is the lowest value if `high` is omitted. random.Generator.integers: which should be used for new code. Examples -------- >>> np.random.randint(2, size=10) array([1, 0, 0, 0, 1, 1, 0, 0, 1, 0]) # random >>> np.random.randint(1, size=10) array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0]) Generate a 2 x 4 array of ints between 0 and 4, inclusive: >>> np.random.randint(5, size=(2, 4)) array([[4, 0, 2, 1], # random [3, 2, 2, 0]]) Generate a 1 x 3 array with 3 different upper bounds >>> np.random.randint(1, [3, 5, 10]) array([2, 2, 9]) # random Generate a 1 by 3 array with 3 different lower bounds >>> np.random.randint([1, 5, 7], 10) array([9, 8, 7]) # random Generate a 2 by 4 array using broadcasting with dtype of uint8 >>> np.random.randint([1, 3, 5, 7], [[10], [20]], dtype=np.uint8) array([[ 8, 6, 9, 7], # random [ 1, 16, 9, 12]], dtype=uint8) tomaxint(size=None) Return a sample of uniformly distributed random integers in the interval [0, ``np.iinfo(np.int_).max``]. The `np.int_` type translates to the C long integer type and its precision is platform dependent. Parameters ---------- size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. Default is None, in which case a single value is returned. Returns ------- out : ndarray Drawn samples, with shape `size`. See Also -------- randint : Uniform sampling over a given half-open interval of integers. random_integers : Uniform sampling over a given closed interval of integers. Examples -------- >>> rs = np.random.RandomState() # need a RandomState object >>> rs.tomaxint((2,2,2)) array([[[1170048599, 1600360186], # random [ 739731006, 1947757578]], [[1871712945, 752307660], [1601631370, 1479324245]]]) >>> rs.tomaxint((2,2,2)) < np.iinfo(np.int_).max array([[[ True, True], [ True, True]], [[ True, True], [ True, True]]]) standard_exponential(size=None) Draw samples from the standard exponential distribution. `standard_exponential` is identical to the exponential distribution with a scale parameter of 1. .. note:: New code should use the `~numpy.random.Generator.standard_exponential` method of a `~numpy.random.Generator` instance instead; please see the :ref:`random-quick-start`. Parameters ---------- size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. Default is None, in which case a single value is returned. Returns ------- out : float or ndarray Drawn samples. See Also -------- random.Generator.standard_exponential: which should be used for new code. Examples -------- Output a 3x8000 array: >>> n = np.random.standard_exponential((3, 8000)) exponential(scale=1.0, size=None) Draw samples from an exponential distribution. Its probability density function is .. math:: f(x; \frac{1}{\beta}) = \frac{1}{\beta} \exp(-\frac{x}{\beta}), for ``x > 0`` and 0 elsewhere. :math:`\beta` is the scale parameter, which is the inverse of the rate parameter :math:`\lambda = 1/\beta`. The rate parameter is an alternative, widely used parameterization of the exponential distribution [3]_. The exponential distribution is a continuous analogue of the geometric distribution. It describes many common situations, such as the size of raindrops measured over many rainstorms [1]_, or the time between page requests to Wikipedia [2]_. .. note:: New code should use the `~numpy.random.Generator.exponential` method of a `~numpy.random.Generator` instance instead; please see the :ref:`random-quick-start`. Parameters ---------- scale : float or array_like of floats The scale parameter, :math:`\beta = 1/\lambda`. Must be non-negative. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``scale`` is a scalar. Otherwise, ``np.array(scale).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized exponential distribution. Examples -------- A real world example: Assume a company has 10000 customer support agents and the average time between customer calls is 4 minutes. >>> n = 10000 >>> time_between_calls = np.random.default_rng().exponential(scale=4, size=n) What is the probability that a customer will call in the next 4 to 5 minutes? >>> x = ((time_between_calls < 5).sum())/n >>> y = ((time_between_calls < 4).sum())/n >>> x-y 0.08 # may vary See Also -------- random.Generator.exponential: which should be used for new code. References ---------- .. [1] Peyton Z. Peebles Jr., "Probability, Random Variables and Random Signal Principles", 4th ed, 2001, p. 57. .. [2] Wikipedia, "Poisson process", https://en.wikipedia.org/wiki/Poisson_process .. [3] Wikipedia, "Exponential distribution", https://en.wikipedia.org/wiki/Exponential_distribution beta(a, b, size=None) Draw samples from a Beta distribution. The Beta distribution is a special case of the Dirichlet distribution, and is related to the Gamma distribution. It has the probability distribution function .. math:: f(x; a,b) = \frac{1}{B(\alpha, \beta)} x^{\alpha - 1} (1 - x)^{\beta - 1}, where the normalization, B, is the beta function, .. math:: B(\alpha, \beta) = \int_0^1 t^{\alpha - 1} (1 - t)^{\beta - 1} dt. It is often seen in Bayesian inference and order statistics. .. note:: New code should use the `~numpy.random.Generator.beta` method of a `~numpy.random.Generator` instance instead; please see the :ref:`random-quick-start`. Parameters ---------- a : float or array_like of floats Alpha, positive (>0). b : float or array_like of floats Beta, positive (>0). size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``a`` and ``b`` are both scalars. Otherwise, ``np.broadcast(a, b).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized beta distribution. See Also -------- random.Generator.beta: which should be used for new code. random(size=None) Return random floats in the half-open interval [0.0, 1.0). Alias for `random_sample` to ease forward-porting to the new random API. random_sample(size=None) Return random floats in the half-open interval [0.0, 1.0). Results are from the "continuous uniform" distribution over the stated interval. To sample :math:`Unif[a, b), b > a` multiply the output of `random_sample` by `(b-a)` and add `a`:: (b - a) * random_sample() + a .. note:: New code should use the `~numpy.random.Generator.random` method of a `~numpy.random.Generator` instance instead; please see the :ref:`random-quick-start`. Parameters ---------- size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. Default is None, in which case a single value is returned. Returns ------- out : float or ndarray of floats Array of random floats of shape `size` (unless ``size=None``, in which case a single float is returned). See Also -------- random.Generator.random: which should be used for new code. Examples -------- >>> np.random.random_sample() 0.47108547995356098 # random >>> type(np.random.random_sample()) >>> np.random.random_sample((5,)) array([ 0.30220482, 0.86820401, 0.1654503 , 0.11659149, 0.54323428]) # random Three-by-two array of random numbers from [-5, 0): >>> 5 * np.random.random_sample((3, 2)) - 5 array([[-3.99149989, -0.52338984], # random [-2.99091858, -0.79479508], [-1.23204345, -1.75224494]]) set_state(state) Set the internal state of the generator from a tuple. For use if one has reason to manually (re-)set the internal state of the bit generator used by the RandomState instance. By default, RandomState uses the "Mersenne Twister"[1]_ pseudo-random number generating algorithm. Parameters ---------- state : {tuple(str, ndarray of 624 uints, int, int, float), dict} The `state` tuple has the following items: 1. the string 'MT19937', specifying the Mersenne Twister algorithm. 2. a 1-D array of 624 unsigned integers ``keys``. 3. an integer ``pos``. 4. an integer ``has_gauss``. 5. a float ``cached_gaussian``. If state is a dictionary, it is directly set using the BitGenerators `state` property. Returns ------- out : None Returns 'None' on success. See Also -------- get_state Notes ----- `set_state` and `get_state` are not needed to work with any of the random distributions in NumPy. If the internal state is manually altered, the user should know exactly what he/she is doing. For backwards compatibility, the form (str, array of 624 uints, int) is also accepted although it is missing some information about the cached Gaussian value: ``state = ('MT19937', keys, pos)``. References ---------- .. [1] M. Matsumoto and T. Nishimura, "Mersenne Twister: A 623-dimensionally equidistributed uniform pseudorandom number generator," *ACM Trans. on Modeling and Computer Simulation*, Vol. 8, No. 1, pp. 3-30, Jan. 1998. get_state(legacy=True) Return a tuple representing the internal state of the generator. For more details, see `set_state`. Parameters ---------- legacy : bool, optional Flag indicating to return a legacy tuple state when the BitGenerator is MT19937, instead of a dict. Raises ValueError if the underlying bit generator is not an instance of MT19937. Returns ------- out : {tuple(str, ndarray of 624 uints, int, int, float), dict} If legacy is True, the returned tuple has the following items: 1. the string 'MT19937'. 2. a 1-D array of 624 unsigned integer keys. 3. an integer ``pos``. 4. an integer ``has_gauss``. 5. a float ``cached_gaussian``. If `legacy` is False, or the BitGenerator is not MT19937, then state is returned as a dictionary. See Also -------- set_state Notes ----- `set_state` and `get_state` are not needed to work with any of the random distributions in NumPy. If the internal state is manually altered, the user should know exactly what he/she is doing. seed(seed=None) Reseed a legacy MT19937 BitGenerator Notes ----- This is a convenience, legacy function. The best practice is to **not** reseed a BitGenerator, rather to recreate a new one. This method is here for legacy reasons. This example demonstrates best practice. >>> from numpy.random import MT19937 >>> from numpy.random import RandomState, SeedSequence >>> rs = RandomState(MT19937(SeedSequence(123456789))) # Later, you want to restart the stream >>> rs = RandomState(MT19937(SeedSequence(987654321))) This function is deprecated. Please call randint({low}, {high} + 1) insteadx must be an integer or at least 1-dimensionalprobabilities are not non-negativenumpy.core.umath failed to importmean and cov must have same lengthlegacy can only be True when the underlyign bitgenerator is an instance of MT19937.get_state and legacy can only be used with the MT19937 BitGenerator. To silence this warning, set `legacy` to False.covariance is not symmetric positive-semidefinite.cov must be 2 dimensional and squarecheck_valid must equal 'warn', 'raise', or 'ignore'can only re-seed a MT19937 BitGeneratora must be 1-dimensional or an integerThis function is deprecated. Please call randint(1, {low} + 1) insteadShuffling a one dimensional array subclass containing objects gives incorrect results for most array subclasses. Please use the new random number API instead: https://numpy.org/doc/stable/reference/random/index.html The new API fixes this issue. This version will not be fixed due to stability guarantees of the API.RandomState.triangular (line 3243)RandomState.standard_t (line 2149)RandomState.standard_normal (line 1384)RandomState.standard_exponential (line 576)RandomState.standard_cauchy (line 2074)RandomState.random_sample (line 384)RandomState.random_integers (line 1288)RandomState.permutation (line 4667)RandomState.noncentral_f (line 1822)RandomState.noncentral_chisquare (line 1985)RandomState.negative_binomial (line 3504)RandomState.multinomial (line 4256)RandomState.exponential (line 499)Providing a dtype with a non-native byteorder is not supported. If you require platform-independent byteorder, call byteswap when required. In future version, providing byteorder will raise a ValueErrorNegative dimensions are not allowedInvalid bit generator. The bit generator must be instantized.Fewer non-zero entries in p than sizeCannot take a larger sample than population when 'replace=False' zipf(a, size=None) Draw samples from a Zipf distribution. Samples are drawn from a Zipf distribution with specified parameter `a` > 1. The Zipf distribution (also known as the zeta distribution) is a discrete probability distribution that satisfies Zipf's law: the frequency of an item is inversely proportional to its rank in a frequency table. .. note:: New code should use the `~numpy.random.Generator.zipf` method of a `~numpy.random.Generator` instance instead; please see the :ref:`random-quick-start`. Parameters ---------- a : float or array_like of floats Distribution parameter. Must be greater than 1. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``a`` is a scalar. Otherwise, ``np.array(a).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized Zipf distribution. See Also -------- scipy.stats.zipf : probability density function, distribution, or cumulative density function, etc. random.Generator.zipf: which should be used for new code. Notes ----- The probability density for the Zipf distribution is .. math:: p(k) = \frac{k^{-a}}{\zeta(a)}, for integers :math:`k \geq 1`, where :math:`\zeta` is the Riemann Zeta function. It is named for the American linguist George Kingsley Zipf, who noted that the frequency of any word in a sample of a language is inversely proportional to its rank in the frequency table. References ---------- .. [1] Zipf, G. K., "Selected Studies of the Principle of Relative Frequency in Language," Cambridge, MA: Harvard Univ. Press, 1932. Examples -------- Draw samples from the distribution: >>> a = 4.0 >>> n = 20000 >>> s = np.random.zipf(a, n) Display the histogram of the samples, along with the expected histogram based on the probability density function: >>> import matplotlib.pyplot as plt >>> from scipy.special import zeta # doctest: +SKIP `bincount` provides a fast histogram for small integers. >>> count = np.bincount(s) >>> k = np.arange(1, s.max() + 1) >>> plt.bar(k, count[1:], alpha=0.5, label='sample count') >>> plt.plot(k, n*(k**-a)/zeta(a), 'k.-', alpha=0.5, ... label='expected count') # doctest: +SKIP >>> plt.semilogy() >>> plt.grid(alpha=0.4) >>> plt.legend() >>> plt.title(f'Zipf sample, a={a}, size={n}') >>> plt.show() weibull(a, size=None) Draw samples from a Weibull distribution. Draw samples from a 1-parameter Weibull distribution with the given shape parameter `a`. .. math:: X = (-ln(U))^{1/a} Here, U is drawn from the uniform distribution over (0,1]. The more common 2-parameter Weibull, including a scale parameter :math:`\lambda` is just :math:`X = \lambda(-ln(U))^{1/a}`. .. note:: New code should use the `~numpy.random.Generator.weibull` method of a `~numpy.random.Generator` instance instead; please see the :ref:`random-quick-start`. Parameters ---------- a : float or array_like of floats Shape parameter of the distribution. Must be nonnegative. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``a`` is a scalar. Otherwise, ``np.array(a).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized Weibull distribution. See Also -------- scipy.stats.weibull_max scipy.stats.weibull_min scipy.stats.genextreme gumbel random.Generator.weibull: which should be used for new code. Notes ----- The Weibull (or Type III asymptotic extreme value distribution for smallest values, SEV Type III, or Rosin-Rammler distribution) is one of a class of Generalized Extreme Value (GEV) distributions used in modeling extreme value problems. This class includes the Gumbel and Frechet distributions. The probability density for the Weibull distribution is .. math:: p(x) = \frac{a} {\lambda}(\frac{x}{\lambda})^{a-1}e^{-(x/\lambda)^a}, where :math:`a` is the shape and :math:`\lambda` the scale. The function has its peak (the mode) at :math:`\lambda(\frac{a-1}{a})^{1/a}`. When ``a = 1``, the Weibull distribution reduces to the exponential distribution. References ---------- .. [1] Waloddi Weibull, Royal Technical University, Stockholm, 1939 "A Statistical Theory Of The Strength Of Materials", Ingeniorsvetenskapsakademiens Handlingar Nr 151, 1939, Generalstabens Litografiska Anstalts Forlag, Stockholm. .. [2] Waloddi Weibull, "A Statistical Distribution Function of Wide Applicability", Journal Of Applied Mechanics ASME Paper 1951. .. [3] Wikipedia, "Weibull distribution", https://en.wikipedia.org/wiki/Weibull_distribution Examples -------- Draw samples from the distribution: >>> a = 5. # shape >>> s = np.random.weibull(a, 1000) Display the histogram of the samples, along with the probability density function: >>> import matplotlib.pyplot as plt >>> x = np.arange(1,100.)/50. >>> def weib(x,n,a): ... return (a / n) * (x / n)**(a - 1) * np.exp(-(x / n)**a) >>> count, bins, ignored = plt.hist(np.random.weibull(5.,1000)) >>> x = np.arange(1,100.)/50. >>> scale = count.max()/weib(x, 1., 5.).max() >>> plt.plot(x, weib(x, 1., 5.)*scale) >>> plt.show() vonmises(mu, kappa, size=None) Draw samples from a von Mises distribution. Samples are drawn from a von Mises distribution with specified mode (mu) and dispersion (kappa), on the interval [-pi, pi]. The von Mises distribution (also known as the circular normal distribution) is a continuous probability distribution on the unit circle. It may be thought of as the circular analogue of the normal distribution. .. note:: New code should use the `~numpy.random.Generator.vonmises` method of a `~numpy.random.Generator` instance instead; please see the :ref:`random-quick-start`. Parameters ---------- mu : float or array_like of floats Mode ("center") of the distribution. kappa : float or array_like of floats Dispersion of the distribution, has to be >=0. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``mu`` and ``kappa`` are both scalars. Otherwise, ``np.broadcast(mu, kappa).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized von Mises distribution. See Also -------- scipy.stats.vonmises : probability density function, distribution, or cumulative density function, etc. random.Generator.vonmises: which should be used for new code. Notes ----- The probability density for the von Mises distribution is .. math:: p(x) = \frac{e^{\kappa cos(x-\mu)}}{2\pi I_0(\kappa)}, where :math:`\mu` is the mode and :math:`\kappa` the dispersion, and :math:`I_0(\kappa)` is the modified Bessel function of order 0. The von Mises is named for Richard Edler von Mises, who was born in Austria-Hungary, in what is now the Ukraine. He fled to the United States in 1939 and became a professor at Harvard. He worked in probability theory, aerodynamics, fluid mechanics, and philosophy of science. References ---------- .. [1] Abramowitz, M. and Stegun, I. A. (Eds.). "Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing," New York: Dover, 1972. .. [2] von Mises, R., "Mathematical Theory of Probability and Statistics", New York: Academic Press, 1964. Examples -------- Draw samples from the distribution: >>> mu, kappa = 0.0, 4.0 # mean and dispersion >>> s = np.random.vonmises(mu, kappa, 1000) Display the histogram of the samples, along with the probability density function: >>> import matplotlib.pyplot as plt >>> from scipy.special import i0 # doctest: +SKIP >>> plt.hist(s, 50, density=True) >>> x = np.linspace(-np.pi, np.pi, num=51) >>> y = np.exp(kappa*np.cos(x-mu))/(2*np.pi*i0(kappa)) # doctest: +SKIP >>> plt.plot(x, y, linewidth=2, color='r') # doctest: +SKIP >>> plt.show() uniform(low=0.0, high=1.0, size=None) Draw samples from a uniform distribution. Samples are uniformly distributed over the half-open interval ``[low, high)`` (includes low, but excludes high). In other words, any value within the given interval is equally likely to be drawn by `uniform`. .. note:: New code should use the `~numpy.random.Generator.uniform` method of a `~numpy.random.Generator` instance instead; please see the :ref:`random-quick-start`. Parameters ---------- low : float or array_like of floats, optional Lower boundary of the output interval. All values generated will be greater than or equal to low. The default value is 0. high : float or array_like of floats Upper boundary of the output interval. All values generated will be less than or equal to high. The high limit may be included in the returned array of floats due to floating-point rounding in the equation ``low + (high-low) * random_sample()``. The default value is 1.0. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``low`` and ``high`` are both scalars. Otherwise, ``np.broadcast(low, high).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized uniform distribution. See Also -------- randint : Discrete uniform distribution, yielding integers. random_integers : Discrete uniform distribution over the closed interval ``[low, high]``. random_sample : Floats uniformly distributed over ``[0, 1)``. random : Alias for `random_sample`. rand : Convenience function that accepts dimensions as input, e.g., ``rand(2,2)`` would generate a 2-by-2 array of floats, uniformly distributed over ``[0, 1)``. random.Generator.uniform: which should be used for new code. Notes ----- The probability density function of the uniform distribution is .. math:: p(x) = \frac{1}{b - a} anywhere within the interval ``[a, b)``, and zero elsewhere. When ``high`` == ``low``, values of ``low`` will be returned. If ``high`` < ``low``, the results are officially undefined and may eventually raise an error, i.e. do not rely on this function to behave when passed arguments satisfying that inequality condition. The ``high`` limit may be included in the returned array of floats due to floating-point rounding in the equation ``low + (high-low) * random_sample()``. For example: >>> x = np.float32(5*0.99999999) >>> x 5.0 Examples -------- Draw samples from the distribution: >>> s = np.random.uniform(-1,0,1000) All values are within the given interval: >>> np.all(s >= -1) True >>> np.all(s < 0) True Display the histogram of the samples, along with the probability density function: >>> import matplotlib.pyplot as plt >>> count, bins, ignored = plt.hist(s, 15, density=True) >>> plt.plot(bins, np.ones_like(bins), linewidth=2, color='r') >>> plt.show() triangular(left, mode, right, size=None) Draw samples from the triangular distribution over the interval ``[left, right]``. The triangular distribution is a continuous probability distribution with lower limit left, peak at mode, and upper limit right. Unlike the other distributions, these parameters directly define the shape of the pdf. .. note:: New code should use the `~numpy.random.Generator.triangular` method of a `~numpy.random.Generator` instance instead; please see the :ref:`random-quick-start`. Parameters ---------- left : float or array_like of floats Lower limit. mode : float or array_like of floats The value where the peak of the distribution occurs. The value must fulfill the condition ``left <= mode <= right``. right : float or array_like of floats Upper limit, must be larger than `left`. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``left``, ``mode``, and ``right`` are all scalars. Otherwise, ``np.broadcast(left, mode, right).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized triangular distribution. See Also -------- random.Generator.triangular: which should be used for new code. Notes ----- The probability density function for the triangular distribution is .. math:: P(x;l, m, r) = \begin{cases} \frac{2(x-l)}{(r-l)(m-l)}& \text{for $l \leq x \leq m$},\\ \frac{2(r-x)}{(r-l)(r-m)}& \text{for $m \leq x \leq r$},\\ 0& \text{otherwise}. \end{cases} The triangular distribution is often used in ill-defined problems where the underlying distribution is not known, but some knowledge of the limits and mode exists. Often it is used in simulations. References ---------- .. [1] Wikipedia, "Triangular distribution" https://en.wikipedia.org/wiki/Triangular_distribution Examples -------- Draw values from the distribution and plot the histogram: >>> import matplotlib.pyplot as plt >>> h = plt.hist(np.random.triangular(-3, 0, 8, 100000), bins=200, ... density=True) >>> plt.show() tomaxint(size=None) Return a sample of uniformly distributed random integers in the interval [0, ``np.iinfo(np.int_).max``]. The `np.int_` type translates to the C long integer type and its precision is platform dependent. Parameters ---------- size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. Default is None, in which case a single value is returned. Returns ------- out : ndarray Drawn samples, with shape `size`. See Also -------- randint : Uniform sampling over a given half-open interval of integers. random_integers : Uniform sampling over a given closed interval of integers. Examples -------- >>> rs = np.random.RandomState() # need a RandomState object >>> rs.tomaxint((2,2,2)) array([[[1170048599, 1600360186], # random [ 739731006, 1947757578]], [[1871712945, 752307660], [1601631370, 1479324245]]]) >>> rs.tomaxint((2,2,2)) < np.iinfo(np.int_).max array([[[ True, True], [ True, True]], [[ True, True], [ True, True]]]) sum(pvals[:-1].astype(np.float64)) > 1.0. The pvals array is cast to 64-bit floating point prior to checking the sum. Precision changes when casting may cause problems even if the sum of the original pvals is valid.state must be a dict or a tuple. standard_t(df, size=None) Draw samples from a standard Student's t distribution with `df` degrees of freedom. A special case of the hyperbolic distribution. As `df` gets large, the result resembles that of the standard normal distribution (`standard_normal`). .. note:: New code should use the `~numpy.random.Generator.standard_t` method of a `~numpy.random.Generator` instance instead; please see the :ref:`random-quick-start`. Parameters ---------- df : float or array_like of floats Degrees of freedom, must be > 0. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``df`` is a scalar. Otherwise, ``np.array(df).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized standard Student's t distribution. See Also -------- random.Generator.standard_t: which should be used for new code. Notes ----- The probability density function for the t distribution is .. math:: P(x, df) = \frac{\Gamma(\frac{df+1}{2})}{\sqrt{\pi df} \Gamma(\frac{df}{2})}\Bigl( 1+\frac{x^2}{df} \Bigr)^{-(df+1)/2} The t test is based on an assumption that the data come from a Normal distribution. The t test provides a way to test whether the sample mean (that is the mean calculated from the data) is a good estimate of the true mean. The derivation of the t-distribution was first published in 1908 by William Gosset while working for the Guinness Brewery in Dublin. Due to proprietary issues, he had to publish under a pseudonym, and so he used the name Student. References ---------- .. [1] Dalgaard, Peter, "Introductory Statistics With R", Springer, 2002. .. [2] Wikipedia, "Student's t-distribution" https://en.wikipedia.org/wiki/Student's_t-distribution Examples -------- From Dalgaard page 83 [1]_, suppose the daily energy intake for 11 women in kilojoules (kJ) is: >>> intake = np.array([5260., 5470, 5640, 6180, 6390, 6515, 6805, 7515, \ ... 7515, 8230, 8770]) Does their energy intake deviate systematically from the recommended value of 7725 kJ? Our null hypothesis will be the absence of deviation, and the alternate hypothesis will be the presence of an effect that could be either positive or negative, hence making our test 2-tailed. Because we are estimating the mean and we have N=11 values in our sample, we have N-1=10 degrees of freedom. We set our significance level to 95% and compute the t statistic using the empirical mean and empirical standard deviation of our intake. We use a ddof of 1 to base the computation of our empirical standard deviation on an unbiased estimate of the variance (note: the final estimate is not unbiased due to the concave nature of the square root). >>> np.mean(intake) 6753.636363636364 >>> intake.std(ddof=1) 1142.1232221373727 >>> t = (np.mean(intake)-7725)/(intake.std(ddof=1)/np.sqrt(len(intake))) >>> t -2.8207540608310198 We draw 1000000 samples from Student's t distribution with the adequate degrees of freedom. >>> import matplotlib.pyplot as plt >>> s = np.random.standard_t(10, size=1000000) >>> h = plt.hist(s, bins=100, density=True) Does our t statistic land in one of the two critical regions found at both tails of the distribution? >>> np.sum(np.abs(t) < np.abs(s)) / float(len(s)) 0.018318 #random < 0.05, statistic is in critical region The probability value for this 2-tailed test is about 1.83%, which is lower than the 5% pre-determined significance threshold. Therefore, the probability of observing values as extreme as our intake conditionally on the null hypothesis being true is too low, and we reject the null hypothesis of no deviation. set_state can only be used with legacy MT19937state instances. rayleigh(scale=1.0, size=None) Draw samples from a Rayleigh distribution. The :math:`\chi` and Weibull distributions are generalizations of the Rayleigh. .. note:: New code should use the `~numpy.random.Generator.rayleigh` method of a `~numpy.random.Generator` instance instead; please see the :ref:`random-quick-start`. Parameters ---------- scale : float or array_like of floats, optional Scale, also equals the mode. Must be non-negative. Default is 1. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``scale`` is a scalar. Otherwise, ``np.array(scale).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized Rayleigh distribution. See Also -------- random.Generator.rayleigh: which should be used for new code. Notes ----- The probability density function for the Rayleigh distribution is .. math:: P(x;scale) = \frac{x}{scale^2}e^{\frac{-x^2}{2 \cdotp scale^2}} The Rayleigh distribution would arise, for example, if the East and North components of the wind velocity had identical zero-mean Gaussian distributions. Then the wind speed would have a Rayleigh distribution. References ---------- .. [1] Brighton Webs Ltd., "Rayleigh Distribution," https://web.archive.org/web/20090514091424/http://brighton-webs.co.uk:80/distributions/rayleigh.asp .. [2] Wikipedia, "Rayleigh distribution" https://en.wikipedia.org/wiki/Rayleigh_distribution Examples -------- Draw values from the distribution and plot the histogram >>> from matplotlib.pyplot import hist >>> values = hist(np.random.rayleigh(3, 100000), bins=200, density=True) Wave heights tend to follow a Rayleigh distribution. If the mean wave height is 1 meter, what fraction of waves are likely to be larger than 3 meters? >>> meanvalue = 1 >>> modevalue = np.sqrt(2 / np.pi) * meanvalue >>> s = np.random.rayleigh(modevalue, 1000000) The percentage of waves larger than 3 meters is: >>> 100.*sum(s>3)/1000000. 0.087300000000000003 # random random_integers(low, high=None, size=None) Random integers of type `np.int_` between `low` and `high`, inclusive. Return random integers of type `np.int_` from the "discrete uniform" distribution in the closed interval [`low`, `high`]. If `high` is None (the default), then results are from [1, `low`]. The `np.int_` type translates to the C long integer type and its precision is platform dependent. This function has been deprecated. Use randint instead. .. deprecated:: 1.11.0 Parameters ---------- low : int Lowest (signed) integer to be drawn from the distribution (unless ``high=None``, in which case this parameter is the *highest* such integer). high : int, optional If provided, the largest (signed) integer to be drawn from the distribution (see above for behavior if ``high=None``). size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. Default is None, in which case a single value is returned. Returns ------- out : int or ndarray of ints `size`-shaped array of random integers from the appropriate distribution, or a single such random int if `size` not provided. See Also -------- randint : Similar to `random_integers`, only for the half-open interval [`low`, `high`), and 0 is the lowest value if `high` is omitted. Notes ----- To sample from N evenly spaced floating-point numbers between a and b, use:: a + (b - a) * (np.random.random_integers(N) - 1) / (N - 1.) Examples -------- >>> np.random.random_integers(5) 4 # random >>> type(np.random.random_integers(5)) >>> np.random.random_integers(5, size=(3,2)) array([[5, 4], # random [3, 3], [4, 5]]) Choose five random numbers from the set of five evenly-spaced numbers between 0 and 2.5, inclusive (*i.e.*, from the set :math:`{0, 5/8, 10/8, 15/8, 20/8}`): >>> 2.5 * (np.random.random_integers(5, size=(5,)) - 1) / 4. array([ 0.625, 1.25 , 0.625, 0.625, 2.5 ]) # random Roll two six sided dice 1000 times and sum the results: >>> d1 = np.random.random_integers(1, 6, 1000) >>> d2 = np.random.random_integers(1, 6, 1000) >>> dsums = d1 + d2 Display results as a histogram: >>> import matplotlib.pyplot as plt >>> count, bins, ignored = plt.hist(dsums, 11, density=True) >>> plt.show() randint(low, high=None, size=None, dtype=int) Return random integers from `low` (inclusive) to `high` (exclusive). Return random integers from the "discrete uniform" distribution of the specified dtype in the "half-open" interval [`low`, `high`). If `high` is None (the default), then results are from [0, `low`). .. note:: New code should use the `~numpy.random.Generator.integers` method of a `~numpy.random.Generator` instance instead; please see the :ref:`random-quick-start`. Parameters ---------- low : int or array-like of ints Lowest (signed) integers to be drawn from the distribution (unless ``high=None``, in which case this parameter is one above the *highest* such integer). high : int or array-like of ints, optional If provided, one above the largest (signed) integer to be drawn from the distribution (see above for behavior if ``high=None``). If array-like, must contain integer values size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. Default is None, in which case a single value is returned. dtype : dtype, optional Desired dtype of the result. Byteorder must be native. The default value is int. .. versionadded:: 1.11.0 Returns ------- out : int or ndarray of ints `size`-shaped array of random integers from the appropriate distribution, or a single such random int if `size` not provided. See Also -------- random_integers : similar to `randint`, only for the closed interval [`low`, `high`], and 1 is the lowest value if `high` is omitted. random.Generator.integers: which should be used for new code. Examples -------- >>> np.random.randint(2, size=10) array([1, 0, 0, 0, 1, 1, 0, 0, 1, 0]) # random >>> np.random.randint(1, size=10) array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0]) Generate a 2 x 4 array of ints between 0 and 4, inclusive: >>> np.random.randint(5, size=(2, 4)) array([[4, 0, 2, 1], # random [3, 2, 2, 0]]) Generate a 1 x 3 array with 3 different upper bounds >>> np.random.randint(1, [3, 5, 10]) array([2, 2, 9]) # random Generate a 1 by 3 array with 3 different lower bounds >>> np.random.randint([1, 5, 7], 10) array([9, 8, 7]) # random Generate a 2 by 4 array using broadcasting with dtype of uint8 >>> np.random.randint([1, 3, 5, 7], [[10], [20]], dtype=np.uint8) array([[ 8, 6, 9, 7], # random [ 1, 16, 9, 12]], dtype=uint8) power(a, size=None) Draws samples in [0, 1] from a power distribution with positive exponent a - 1. Also known as the power function distribution. .. note:: New code should use the `~numpy.random.Generator.power` method of a `~numpy.random.Generator` instance instead; please see the :ref:`random-quick-start`. Parameters ---------- a : float or array_like of floats Parameter of the distribution. Must be non-negative. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``a`` is a scalar. Otherwise, ``np.array(a).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized power distribution. Raises ------ ValueError If a <= 0. See Also -------- random.Generator.power: which should be used for new code. Notes ----- The probability density function is .. math:: P(x; a) = ax^{a-1}, 0 \le x \le 1, a>0. The power function distribution is just the inverse of the Pareto distribution. It may also be seen as a special case of the Beta distribution. It is used, for example, in modeling the over-reporting of insurance claims. References ---------- .. [1] Christian Kleiber, Samuel Kotz, "Statistical size distributions in economics and actuarial sciences", Wiley, 2003. .. [2] Heckert, N. A. and Filliben, James J. "NIST Handbook 148: Dataplot Reference Manual, Volume 2: Let Subcommands and Library Functions", National Institute of Standards and Technology Handbook Series, June 2003. https://www.itl.nist.gov/div898/software/dataplot/refman2/auxillar/powpdf.pdf Examples -------- Draw samples from the distribution: >>> a = 5. # shape >>> samples = 1000 >>> s = np.random.power(a, samples) Display the histogram of the samples, along with the probability density function: >>> import matplotlib.pyplot as plt >>> count, bins, ignored = plt.hist(s, bins=30) >>> x = np.linspace(0, 1, 100) >>> y = a*x**(a-1.) >>> normed_y = samples*np.diff(bins)[0]*y >>> plt.plot(x, normed_y) >>> plt.show() Compare the power function distribution to the inverse of the Pareto. >>> from scipy import stats # doctest: +SKIP >>> rvs = np.random.power(5, 1000000) >>> rvsp = np.random.pareto(5, 1000000) >>> xx = np.linspace(0,1,100) >>> powpdf = stats.powerlaw.pdf(xx,5) # doctest: +SKIP >>> plt.figure() >>> plt.hist(rvs, bins=50, density=True) >>> plt.plot(xx,powpdf,'r-') # doctest: +SKIP >>> plt.title('np.random.power(5)') >>> plt.figure() >>> plt.hist(1./(1.+rvsp), bins=50, density=True) >>> plt.plot(xx,powpdf,'r-') # doctest: +SKIP >>> plt.title('inverse of 1 + np.random.pareto(5)') >>> plt.figure() >>> plt.hist(1./(1.+rvsp), bins=50, density=True) >>> plt.plot(xx,powpdf,'r-') # doctest: +SKIP >>> plt.title('inverse of stats.pareto(5)') pareto(a, size=None) Draw samples from a Pareto II or Lomax distribution with specified shape. The Lomax or Pareto II distribution is a shifted Pareto distribution. The classical Pareto distribution can be obtained from the Lomax distribution by adding 1 and multiplying by the scale parameter ``m`` (see Notes). The smallest value of the Lomax distribution is zero while for the classical Pareto distribution it is ``mu``, where the standard Pareto distribution has location ``mu = 1``. Lomax can also be considered as a simplified version of the Generalized Pareto distribution (available in SciPy), with the scale set to one and the location set to zero. The Pareto distribution must be greater than zero, and is unbounded above. It is also known as the "80-20 rule". In this distribution, 80 percent of the weights are in the lowest 20 percent of the range, while the other 20 percent fill the remaining 80 percent of the range. .. note:: New code should use the `~numpy.random.Generator.pareto` method of a `~numpy.random.Generator` instance instead; please see the :ref:`random-quick-start`. Parameters ---------- a : float or array_like of floats Shape of the distribution. Must be positive. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``a`` is a scalar. Otherwise, ``np.array(a).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized Pareto distribution. See Also -------- scipy.stats.lomax : probability density function, distribution or cumulative density function, etc. scipy.stats.genpareto : probability density function, distribution or cumulative density function, etc. random.Generator.pareto: which should be used for new code. Notes ----- The probability density for the Pareto distribution is .. math:: p(x) = \frac{am^a}{x^{a+1}} where :math:`a` is the shape and :math:`m` the scale. The Pareto distribution, named after the Italian economist Vilfredo Pareto, is a power law probability distribution useful in many real world problems. Outside the field of economics it is generally referred to as the Bradford distribution. Pareto developed the distribution to describe the distribution of wealth in an economy. It has also found use in insurance, web page access statistics, oil field sizes, and many other problems, including the download frequency for projects in Sourceforge [1]_. It is one of the so-called "fat-tailed" distributions. References ---------- .. [1] Francis Hunt and Paul Johnson, On the Pareto Distribution of Sourceforge projects. .. [2] Pareto, V. (1896). Course of Political Economy. Lausanne. .. [3] Reiss, R.D., Thomas, M.(2001), Statistical Analysis of Extreme Values, Birkhauser Verlag, Basel, pp 23-30. .. [4] Wikipedia, "Pareto distribution", https://en.wikipedia.org/wiki/Pareto_distribution Examples -------- Draw samples from the distribution: >>> a, m = 3., 2. # shape and mode >>> s = (np.random.pareto(a, 1000) + 1) * m Display the histogram of the samples, along with the probability density function: >>> import matplotlib.pyplot as plt >>> count, bins, _ = plt.hist(s, 100, density=True) >>> fit = a*m**a / bins**(a+1) >>> plt.plot(bins, max(count)*fit/max(fit), linewidth=2, color='r') >>> plt.show() numpy.core.multiarray failed to import normal(loc=0.0, scale=1.0, size=None) Draw random samples from a normal (Gaussian) distribution. The probability density function of the normal distribution, first derived by De Moivre and 200 years later by both Gauss and Laplace independently [2]_, is often called the bell curve because of its characteristic shape (see the example below). The normal distributions occurs often in nature. For example, it describes the commonly occurring distribution of samples influenced by a large number of tiny, random disturbances, each with its own unique distribution [2]_. .. note:: New code should use the `~numpy.random.Generator.normal` method of a `~numpy.random.Generator` instance instead; please see the :ref:`random-quick-start`. Parameters ---------- loc : float or array_like of floats Mean ("centre") of the distribution. scale : float or array_like of floats Standard deviation (spread or "width") of the distribution. Must be non-negative. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``loc`` and ``scale`` are both scalars. Otherwise, ``np.broadcast(loc, scale).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized normal distribution. See Also -------- scipy.stats.norm : probability density function, distribution or cumulative density function, etc. random.Generator.normal: which should be used for new code. Notes ----- The probability density for the Gaussian distribution is .. math:: p(x) = \frac{1}{\sqrt{ 2 \pi \sigma^2 }} e^{ - \frac{ (x - \mu)^2 } {2 \sigma^2} }, where :math:`\mu` is the mean and :math:`\sigma` the standard deviation. The square of the standard deviation, :math:`\sigma^2`, is called the variance. The function has its peak at the mean, and its "spread" increases with the standard deviation (the function reaches 0.607 times its maximum at :math:`x + \sigma` and :math:`x - \sigma` [2]_). This implies that normal is more likely to return samples lying close to the mean, rather than those far away. References ---------- .. [1] Wikipedia, "Normal distribution", https://en.wikipedia.org/wiki/Normal_distribution .. [2] P. R. Peebles Jr., "Central Limit Theorem" in "Probability, Random Variables and Random Signal Principles", 4th ed., 2001, pp. 51, 51, 125. Examples -------- Draw samples from the distribution: >>> mu, sigma = 0, 0.1 # mean and standard deviation >>> s = np.random.normal(mu, sigma, 1000) Verify the mean and the variance: >>> abs(mu - np.mean(s)) 0.0 # may vary >>> abs(sigma - np.std(s, ddof=1)) 0.1 # may vary Display the histogram of the samples, along with the probability density function: >>> import matplotlib.pyplot as plt >>> count, bins, ignored = plt.hist(s, 30, density=True) >>> plt.plot(bins, 1/(sigma * np.sqrt(2 * np.pi)) * ... np.exp( - (bins - mu)**2 / (2 * sigma**2) ), ... linewidth=2, color='r') >>> plt.show() Two-by-four array of samples from the normal distribution with mean 3 and standard deviation 2.5: >>> np.random.normal(3, 2.5, size=(2, 4)) array([[-4.49401501, 4.00950034, -1.81814867, 7.29718677], # random [ 0.39924804, 4.68456316, 4.99394529, 4.84057254]]) # random noncentral_f(dfnum, dfden, nonc, size=None) Draw samples from the noncentral F distribution. Samples are drawn from an F distribution with specified parameters, `dfnum` (degrees of freedom in numerator) and `dfden` (degrees of freedom in denominator), where both parameters > 1. `nonc` is the non-centrality parameter. .. note:: New code should use the `~numpy.random.Generator.noncentral_f` method of a `~numpy.random.Generator` instance instead; please see the :ref:`random-quick-start`. Parameters ---------- dfnum : float or array_like of floats Numerator degrees of freedom, must be > 0. .. versionchanged:: 1.14.0 Earlier NumPy versions required dfnum > 1. dfden : float or array_like of floats Denominator degrees of freedom, must be > 0. nonc : float or array_like of floats Non-centrality parameter, the sum of the squares of the numerator means, must be >= 0. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``dfnum``, ``dfden``, and ``nonc`` are all scalars. Otherwise, ``np.broadcast(dfnum, dfden, nonc).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized noncentral Fisher distribution. See Also -------- random.Generator.noncentral_f: which should be used for new code. Notes ----- When calculating the power of an experiment (power = probability of rejecting the null hypothesis when a specific alternative is true) the non-central F statistic becomes important. When the null hypothesis is true, the F statistic follows a central F distribution. When the null hypothesis is not true, then it follows a non-central F statistic. References ---------- .. [1] Weisstein, Eric W. "Noncentral F-Distribution." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/NoncentralF-Distribution.html .. [2] Wikipedia, "Noncentral F-distribution", https://en.wikipedia.org/wiki/Noncentral_F-distribution Examples -------- In a study, testing for a specific alternative to the null hypothesis requires use of the Noncentral F distribution. We need to calculate the area in the tail of the distribution that exceeds the value of the F distribution for the null hypothesis. We'll plot the two probability distributions for comparison. >>> dfnum = 3 # between group deg of freedom >>> dfden = 20 # within groups degrees of freedom >>> nonc = 3.0 >>> nc_vals = np.random.noncentral_f(dfnum, dfden, nonc, 1000000) >>> NF = np.histogram(nc_vals, bins=50, density=True) >>> c_vals = np.random.f(dfnum, dfden, 1000000) >>> F = np.histogram(c_vals, bins=50, density=True) >>> import matplotlib.pyplot as plt >>> plt.plot(F[1][1:], F[0]) >>> plt.plot(NF[1][1:], NF[0]) >>> plt.show() noncentral_chisquare(df, nonc, size=None) Draw samples from a noncentral chi-square distribution. The noncentral :math:`\chi^2` distribution is a generalization of the :math:`\chi^2` distribution. .. note:: New code should use the `~numpy.random.Generator.noncentral_chisquare` method of a `~numpy.random.Generator` instance instead; please see the :ref:`random-quick-start`. Parameters ---------- df : float or array_like of floats Degrees of freedom, must be > 0. .. versionchanged:: 1.10.0 Earlier NumPy versions required dfnum > 1. nonc : float or array_like of floats Non-centrality, must be non-negative. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``df`` and ``nonc`` are both scalars. Otherwise, ``np.broadcast(df, nonc).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized noncentral chi-square distribution. See Also -------- random.Generator.noncentral_chisquare: which should be used for new code. Notes ----- The probability density function for the noncentral Chi-square distribution is .. math:: P(x;df,nonc) = \sum^{\infty}_{i=0} \frac{e^{-nonc/2}(nonc/2)^{i}}{i!} P_{Y_{df+2i}}(x), where :math:`Y_{q}` is the Chi-square with q degrees of freedom. References ---------- .. [1] Wikipedia, "Noncentral chi-squared distribution" https://en.wikipedia.org/wiki/Noncentral_chi-squared_distribution Examples -------- Draw values from the distribution and plot the histogram >>> import matplotlib.pyplot as plt >>> values = plt.hist(np.random.noncentral_chisquare(3, 20, 100000), ... bins=200, density=True) >>> plt.show() Draw values from a noncentral chisquare with very small noncentrality, and compare to a chisquare. >>> plt.figure() >>> values = plt.hist(np.random.noncentral_chisquare(3, .0000001, 100000), ... bins=np.arange(0., 25, .1), density=True) >>> values2 = plt.hist(np.random.chisquare(3, 100000), ... bins=np.arange(0., 25, .1), density=True) >>> plt.plot(values[1][0:-1], values[0]-values2[0], 'ob') >>> plt.show() Demonstrate how large values of non-centrality lead to a more symmetric distribution. >>> plt.figure() >>> values = plt.hist(np.random.noncentral_chisquare(3, 20, 100000), ... bins=200, density=True) >>> plt.show() negative_binomial(n, p, size=None) Draw samples from a negative binomial distribution. Samples are drawn from a negative binomial distribution with specified parameters, `n` successes and `p` probability of success where `n` is > 0 and `p` is in the interval [0, 1]. .. note:: New code should use the `~numpy.random.Generator.negative_binomial` method of a `~numpy.random.Generator` instance instead; please see the :ref:`random-quick-start`. Parameters ---------- n : float or array_like of floats Parameter of the distribution, > 0. p : float or array_like of floats Parameter of the distribution, >= 0 and <=1. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``n`` and ``p`` are both scalars. Otherwise, ``np.broadcast(n, p).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized negative binomial distribution, where each sample is equal to N, the number of failures that occurred before a total of n successes was reached. See Also -------- random.Generator.negative_binomial: which should be used for new code. Notes ----- The probability mass function of the negative binomial distribution is .. math:: P(N;n,p) = \frac{\Gamma(N+n)}{N!\Gamma(n)}p^{n}(1-p)^{N}, where :math:`n` is the number of successes, :math:`p` is the probability of success, :math:`N+n` is the number of trials, and :math:`\Gamma` is the gamma function. When :math:`n` is an integer, :math:`\frac{\Gamma(N+n)}{N!\Gamma(n)} = \binom{N+n-1}{N}`, which is the more common form of this term in the pmf. The negative binomial distribution gives the probability of N failures given n successes, with a success on the last trial. If one throws a die repeatedly until the third time a "1" appears, then the probability distribution of the number of non-"1"s that appear before the third "1" is a negative binomial distribution. References ---------- .. [1] Weisstein, Eric W. "Negative Binomial Distribution." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/NegativeBinomialDistribution.html .. [2] Wikipedia, "Negative binomial distribution", https://en.wikipedia.org/wiki/Negative_binomial_distribution Examples -------- Draw samples from the distribution: A real world example. A company drills wild-cat oil exploration wells, each with an estimated probability of success of 0.1. What is the probability of having one success for each successive well, that is what is the probability of a single success after drilling 5 wells, after 6 wells, etc.? >>> s = np.random.negative_binomial(1, 0.1, 100000) >>> for i in range(1, 11): # doctest: +SKIP ... probability = sum(s>> mean = [0, 0] >>> cov = [[1, 0], [0, 100]] # diagonal covariance Diagonal covariance means that points are oriented along x or y-axis: >>> import matplotlib.pyplot as plt >>> x, y = np.random.multivariate_normal(mean, cov, 5000).T >>> plt.plot(x, y, 'x') >>> plt.axis('equal') >>> plt.show() Note that the covariance matrix must be positive semidefinite (a.k.a. nonnegative-definite). Otherwise, the behavior of this method is undefined and backwards compatibility is not guaranteed. References ---------- .. [1] Papoulis, A., "Probability, Random Variables, and Stochastic Processes," 3rd ed., New York: McGraw-Hill, 1991. .. [2] Duda, R. O., Hart, P. E., and Stork, D. G., "Pattern Classification," 2nd ed., New York: Wiley, 2001. Examples -------- >>> mean = (1, 2) >>> cov = [[1, 0], [0, 1]] >>> x = np.random.multivariate_normal(mean, cov, (3, 3)) >>> x.shape (3, 3, 2) Here we generate 800 samples from the bivariate normal distribution with mean [0, 0] and covariance matrix [[6, -3], [-3, 3.5]]. The expected variances of the first and second components of the sample are 6 and 3.5, respectively, and the expected correlation coefficient is -3/sqrt(6*3.5) ≈ -0.65465. >>> cov = np.array([[6, -3], [-3, 3.5]]) >>> pts = np.random.multivariate_normal([0, 0], cov, size=800) Check that the mean, covariance, and correlation coefficient of the sample are close to the expected values: >>> pts.mean(axis=0) array([ 0.0326911 , -0.01280782]) # may vary >>> np.cov(pts.T) array([[ 5.96202397, -2.85602287], [-2.85602287, 3.47613949]]) # may vary >>> np.corrcoef(pts.T)[0, 1] -0.6273591314603949 # may vary We can visualize this data with a scatter plot. The orientation of the point cloud illustrates the negative correlation of the components of this sample. >>> import matplotlib.pyplot as plt >>> plt.plot(pts[:, 0], pts[:, 1], '.', alpha=0.5) >>> plt.axis('equal') >>> plt.grid() >>> plt.show() multinomial(n, pvals, size=None) Draw samples from a multinomial distribution. The multinomial distribution is a multivariate generalization of the binomial distribution. Take an experiment with one of ``p`` possible outcomes. An example of such an experiment is throwing a dice, where the outcome can be 1 through 6. Each sample drawn from the distribution represents `n` such experiments. Its values, ``X_i = [X_0, X_1, ..., X_p]``, represent the number of times the outcome was ``i``. .. note:: New code should use the `~numpy.random.Generator.multinomial` method of a `~numpy.random.Generator` instance instead; please see the :ref:`random-quick-start`. Parameters ---------- n : int Number of experiments. pvals : sequence of floats, length p Probabilities of each of the ``p`` different outcomes. These must sum to 1 (however, the last element is always assumed to account for the remaining probability, as long as ``sum(pvals[:-1]) <= 1)``. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. Default is None, in which case a single value is returned. Returns ------- out : ndarray The drawn samples, of shape *size*, if that was provided. If not, the shape is ``(N,)``. In other words, each entry ``out[i,j,...,:]`` is an N-dimensional value drawn from the distribution. See Also -------- random.Generator.multinomial: which should be used for new code. Examples -------- Throw a dice 20 times: >>> np.random.multinomial(20, [1/6.]*6, size=1) array([[4, 1, 7, 5, 2, 1]]) # random It landed 4 times on 1, once on 2, etc. Now, throw the dice 20 times, and 20 times again: >>> np.random.multinomial(20, [1/6.]*6, size=2) array([[3, 4, 3, 3, 4, 3], # random [2, 4, 3, 4, 0, 7]]) For the first run, we threw 3 times 1, 4 times 2, etc. For the second, we threw 2 times 1, 4 times 2, etc. A loaded die is more likely to land on number 6: >>> np.random.multinomial(100, [1/7.]*5 + [2/7.]) array([11, 16, 14, 17, 16, 26]) # random The probability inputs should be normalized. As an implementation detail, the value of the last entry is ignored and assumed to take up any leftover probability mass, but this should not be relied on. A biased coin which has twice as much weight on one side as on the other should be sampled like so: >>> np.random.multinomial(100, [1.0 / 3, 2.0 / 3]) # RIGHT array([38, 62]) # random not like: >>> np.random.multinomial(100, [1.0, 2.0]) # WRONG Traceback (most recent call last): ValueError: pvals < 0, pvals > 1 or pvals contains NaNs logseries(p, size=None) Draw samples from a logarithmic series distribution. Samples are drawn from a log series distribution with specified shape parameter, 0 <= ``p`` < 1. .. note:: New code should use the `~numpy.random.Generator.logseries` method of a `~numpy.random.Generator` instance instead; please see the :ref:`random-quick-start`. Parameters ---------- p : float or array_like of floats Shape parameter for the distribution. Must be in the range [0, 1). size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``p`` is a scalar. Otherwise, ``np.array(p).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized logarithmic series distribution. See Also -------- scipy.stats.logser : probability density function, distribution or cumulative density function, etc. random.Generator.logseries: which should be used for new code. Notes ----- The probability density for the Log Series distribution is .. math:: P(k) = \frac{-p^k}{k \ln(1-p)}, where p = probability. The log series distribution is frequently used to represent species richness and occurrence, first proposed by Fisher, Corbet, and Williams in 1943 [2]. It may also be used to model the numbers of occupants seen in cars [3]. References ---------- .. [1] Buzas, Martin A.; Culver, Stephen J., Understanding regional species diversity through the log series distribution of occurrences: BIODIVERSITY RESEARCH Diversity & Distributions, Volume 5, Number 5, September 1999 , pp. 187-195(9). .. [2] Fisher, R.A,, A.S. Corbet, and C.B. Williams. 1943. The relation between the number of species and the number of individuals in a random sample of an animal population. Journal of Animal Ecology, 12:42-58. .. [3] D. J. Hand, F. Daly, D. Lunn, E. Ostrowski, A Handbook of Small Data Sets, CRC Press, 1994. .. [4] Wikipedia, "Logarithmic distribution", https://en.wikipedia.org/wiki/Logarithmic_distribution Examples -------- Draw samples from the distribution: >>> a = .6 >>> s = np.random.logseries(a, 10000) >>> import matplotlib.pyplot as plt >>> count, bins, ignored = plt.hist(s) # plot against distribution >>> def logseries(k, p): ... return -p**k/(k*np.log(1-p)) >>> plt.plot(bins, logseries(bins, a)*count.max()/ ... logseries(bins, a).max(), 'r') >>> plt.show() lognormal(mean=0.0, sigma=1.0, size=None) Draw samples from a log-normal distribution. Draw samples from a log-normal distribution with specified mean, standard deviation, and array shape. Note that the mean and standard deviation are not the values for the distribution itself, but of the underlying normal distribution it is derived from. .. note:: New code should use the `~numpy.random.Generator.lognormal` method of a `~numpy.random.Generator` instance instead; please see the :ref:`random-quick-start`. Parameters ---------- mean : float or array_like of floats, optional Mean value of the underlying normal distribution. Default is 0. sigma : float or array_like of floats, optional Standard deviation of the underlying normal distribution. Must be non-negative. Default is 1. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``mean`` and ``sigma`` are both scalars. Otherwise, ``np.broadcast(mean, sigma).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized log-normal distribution. See Also -------- scipy.stats.lognorm : probability density function, distribution, cumulative density function, etc. random.Generator.lognormal: which should be used for new code. Notes ----- A variable `x` has a log-normal distribution if `log(x)` is normally distributed. The probability density function for the log-normal distribution is: .. math:: p(x) = \frac{1}{\sigma x \sqrt{2\pi}} e^{(-\frac{(ln(x)-\mu)^2}{2\sigma^2})} where :math:`\mu` is the mean and :math:`\sigma` is the standard deviation of the normally distributed logarithm of the variable. A log-normal distribution results if a random variable is the *product* of a large number of independent, identically-distributed variables in the same way that a normal distribution results if the variable is the *sum* of a large number of independent, identically-distributed variables. References ---------- .. [1] Limpert, E., Stahel, W. A., and Abbt, M., "Log-normal Distributions across the Sciences: Keys and Clues," BioScience, Vol. 51, No. 5, May, 2001. https://stat.ethz.ch/~stahel/lognormal/bioscience.pdf .. [2] Reiss, R.D. and Thomas, M., "Statistical Analysis of Extreme Values," Basel: Birkhauser Verlag, 2001, pp. 31-32. Examples -------- Draw samples from the distribution: >>> mu, sigma = 3., 1. # mean and standard deviation >>> s = np.random.lognormal(mu, sigma, 1000) Display the histogram of the samples, along with the probability density function: >>> import matplotlib.pyplot as plt >>> count, bins, ignored = plt.hist(s, 100, density=True, align='mid') >>> x = np.linspace(min(bins), max(bins), 10000) >>> pdf = (np.exp(-(np.log(x) - mu)**2 / (2 * sigma**2)) ... / (x * sigma * np.sqrt(2 * np.pi))) >>> plt.plot(x, pdf, linewidth=2, color='r') >>> plt.axis('tight') >>> plt.show() Demonstrate that taking the products of random samples from a uniform distribution can be fit well by a log-normal probability density function. >>> # Generate a thousand samples: each is the product of 100 random >>> # values, drawn from a normal distribution. >>> b = [] >>> for i in range(1000): ... a = 10. + np.random.standard_normal(100) ... b.append(np.prod(a)) >>> b = np.array(b) / np.min(b) # scale values to be positive >>> count, bins, ignored = plt.hist(b, 100, density=True, align='mid') >>> sigma = np.std(np.log(b)) >>> mu = np.mean(np.log(b)) >>> x = np.linspace(min(bins), max(bins), 10000) >>> pdf = (np.exp(-(np.log(x) - mu)**2 / (2 * sigma**2)) ... / (x * sigma * np.sqrt(2 * np.pi))) >>> plt.plot(x, pdf, color='r', linewidth=2) >>> plt.show() logistic(loc=0.0, scale=1.0, size=None) Draw samples from a logistic distribution. Samples are drawn from a logistic distribution with specified parameters, loc (location or mean, also median), and scale (>0). .. note:: New code should use the `~numpy.random.Generator.logistic` method of a `~numpy.random.Generator` instance instead; please see the :ref:`random-quick-start`. Parameters ---------- loc : float or array_like of floats, optional Parameter of the distribution. Default is 0. scale : float or array_like of floats, optional Parameter of the distribution. Must be non-negative. Default is 1. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``loc`` and ``scale`` are both scalars. Otherwise, ``np.broadcast(loc, scale).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized logistic distribution. See Also -------- scipy.stats.logistic : probability density function, distribution or cumulative density function, etc. random.Generator.logistic: which should be used for new code. Notes ----- The probability density for the Logistic distribution is .. math:: P(x) = P(x) = \frac{e^{-(x-\mu)/s}}{s(1+e^{-(x-\mu)/s})^2}, where :math:`\mu` = location and :math:`s` = scale. The Logistic distribution is used in Extreme Value problems where it can act as a mixture of Gumbel distributions, in Epidemiology, and by the World Chess Federation (FIDE) where it is used in the Elo ranking system, assuming the performance of each player is a logistically distributed random variable. References ---------- .. [1] Reiss, R.-D. and Thomas M. (2001), "Statistical Analysis of Extreme Values, from Insurance, Finance, Hydrology and Other Fields," Birkhauser Verlag, Basel, pp 132-133. .. [2] Weisstein, Eric W. "Logistic Distribution." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/LogisticDistribution.html .. [3] Wikipedia, "Logistic-distribution", https://en.wikipedia.org/wiki/Logistic_distribution Examples -------- Draw samples from the distribution: >>> loc, scale = 10, 1 >>> s = np.random.logistic(loc, scale, 10000) >>> import matplotlib.pyplot as plt >>> count, bins, ignored = plt.hist(s, bins=50) # plot against distribution >>> def logist(x, loc, scale): ... return np.exp((loc-x)/scale)/(scale*(1+np.exp((loc-x)/scale))**2) >>> lgst_val = logist(bins, loc, scale) >>> plt.plot(bins, lgst_val * count.max() / lgst_val.max()) >>> plt.show() hypergeometric(ngood, nbad, nsample, size=None) Draw samples from a Hypergeometric distribution. Samples are drawn from a hypergeometric distribution with specified parameters, `ngood` (ways to make a good selection), `nbad` (ways to make a bad selection), and `nsample` (number of items sampled, which is less than or equal to the sum ``ngood + nbad``). .. note:: New code should use the `~numpy.random.Generator.hypergeometric` method of a `~numpy.random.Generator` instance instead; please see the :ref:`random-quick-start`. Parameters ---------- ngood : int or array_like of ints Number of ways to make a good selection. Must be nonnegative. nbad : int or array_like of ints Number of ways to make a bad selection. Must be nonnegative. nsample : int or array_like of ints Number of items sampled. Must be at least 1 and at most ``ngood + nbad``. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if `ngood`, `nbad`, and `nsample` are all scalars. Otherwise, ``np.broadcast(ngood, nbad, nsample).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized hypergeometric distribution. Each sample is the number of good items within a randomly selected subset of size `nsample` taken from a set of `ngood` good items and `nbad` bad items. See Also -------- scipy.stats.hypergeom : probability density function, distribution or cumulative density function, etc. random.Generator.hypergeometric: which should be used for new code. Notes ----- The probability density for the Hypergeometric distribution is .. math:: P(x) = \frac{\binom{g}{x}\binom{b}{n-x}}{\binom{g+b}{n}}, where :math:`0 \le x \le n` and :math:`n-b \le x \le g` for P(x) the probability of ``x`` good results in the drawn sample, g = `ngood`, b = `nbad`, and n = `nsample`. Consider an urn with black and white marbles in it, `ngood` of them are black and `nbad` are white. If you draw `nsample` balls without replacement, then the hypergeometric distribution describes the distribution of black balls in the drawn sample. Note that this distribution is very similar to the binomial distribution, except that in this case, samples are drawn without replacement, whereas in the Binomial case samples are drawn with replacement (or the sample space is infinite). As the sample space becomes large, this distribution approaches the binomial. References ---------- .. [1] Lentner, Marvin, "Elementary Applied Statistics", Bogden and Quigley, 1972. .. [2] Weisstein, Eric W. "Hypergeometric Distribution." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/HypergeometricDistribution.html .. [3] Wikipedia, "Hypergeometric distribution", https://en.wikipedia.org/wiki/Hypergeometric_distribution Examples -------- Draw samples from the distribution: >>> ngood, nbad, nsamp = 100, 2, 10 # number of good, number of bad, and number of samples >>> s = np.random.hypergeometric(ngood, nbad, nsamp, 1000) >>> from matplotlib.pyplot import hist >>> hist(s) # note that it is very unlikely to grab both bad items Suppose you have an urn with 15 white and 15 black marbles. If you pull 15 marbles at random, how likely is it that 12 or more of them are one color? >>> s = np.random.hypergeometric(15, 15, 15, 100000) >>> sum(s>=12)/100000. + sum(s<=3)/100000. # answer = 0.003 ... pretty unlikely! gumbel(loc=0.0, scale=1.0, size=None) Draw samples from a Gumbel distribution. Draw samples from a Gumbel distribution with specified location and scale. For more information on the Gumbel distribution, see Notes and References below. .. note:: New code should use the `~numpy.random.Generator.gumbel` method of a `~numpy.random.Generator` instance instead; please see the :ref:`random-quick-start`. Parameters ---------- loc : float or array_like of floats, optional The location of the mode of the distribution. Default is 0. scale : float or array_like of floats, optional The scale parameter of the distribution. Default is 1. Must be non- negative. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``loc`` and ``scale`` are both scalars. Otherwise, ``np.broadcast(loc, scale).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized Gumbel distribution. See Also -------- scipy.stats.gumbel_l scipy.stats.gumbel_r scipy.stats.genextreme weibull random.Generator.gumbel: which should be used for new code. Notes ----- The Gumbel (or Smallest Extreme Value (SEV) or the Smallest Extreme Value Type I) distribution is one of a class of Generalized Extreme Value (GEV) distributions used in modeling extreme value problems. The Gumbel is a special case of the Extreme Value Type I distribution for maximums from distributions with "exponential-like" tails. The probability density for the Gumbel distribution is .. math:: p(x) = \frac{e^{-(x - \mu)/ \beta}}{\beta} e^{ -e^{-(x - \mu)/ \beta}}, where :math:`\mu` is the mode, a location parameter, and :math:`\beta` is the scale parameter. The Gumbel (named for German mathematician Emil Julius Gumbel) was used very early in the hydrology literature, for modeling the occurrence of flood events. It is also used for modeling maximum wind speed and rainfall rates. It is a "fat-tailed" distribution - the probability of an event in the tail of the distribution is larger than if one used a Gaussian, hence the surprisingly frequent occurrence of 100-year floods. Floods were initially modeled as a Gaussian process, which underestimated the frequency of extreme events. It is one of a class of extreme value distributions, the Generalized Extreme Value (GEV) distributions, which also includes the Weibull and Frechet. The function has a mean of :math:`\mu + 0.57721\beta` and a variance of :math:`\frac{\pi^2}{6}\beta^2`. References ---------- .. [1] Gumbel, E. J., "Statistics of Extremes," New York: Columbia University Press, 1958. .. [2] Reiss, R.-D. and Thomas, M., "Statistical Analysis of Extreme Values from Insurance, Finance, Hydrology and Other Fields," Basel: Birkhauser Verlag, 2001. Examples -------- Draw samples from the distribution: >>> mu, beta = 0, 0.1 # location and scale >>> s = np.random.gumbel(mu, beta, 1000) Display the histogram of the samples, along with the probability density function: >>> import matplotlib.pyplot as plt >>> count, bins, ignored = plt.hist(s, 30, density=True) >>> plt.plot(bins, (1/beta)*np.exp(-(bins - mu)/beta) ... * np.exp( -np.exp( -(bins - mu) /beta) ), ... linewidth=2, color='r') >>> plt.show() Show how an extreme value distribution can arise from a Gaussian process and compare to a Gaussian: >>> means = [] >>> maxima = [] >>> for i in range(0,1000) : ... a = np.random.normal(mu, beta, 1000) ... means.append(a.mean()) ... maxima.append(a.max()) >>> count, bins, ignored = plt.hist(maxima, 30, density=True) >>> beta = np.std(maxima) * np.sqrt(6) / np.pi >>> mu = np.mean(maxima) - 0.57721*beta >>> plt.plot(bins, (1/beta)*np.exp(-(bins - mu)/beta) ... * np.exp(-np.exp(-(bins - mu)/beta)), ... linewidth=2, color='r') >>> plt.plot(bins, 1/(beta * np.sqrt(2 * np.pi)) ... * np.exp(-(bins - mu)**2 / (2 * beta**2)), ... linewidth=2, color='g') >>> plt.show() geometric(p, size=None) Draw samples from the geometric distribution. Bernoulli trials are experiments with one of two outcomes: success or failure (an example of such an experiment is flipping a coin). The geometric distribution models the number of trials that must be run in order to achieve success. It is therefore supported on the positive integers, ``k = 1, 2, ...``. The probability mass function of the geometric distribution is .. math:: f(k) = (1 - p)^{k - 1} p where `p` is the probability of success of an individual trial. .. note:: New code should use the `~numpy.random.Generator.geometric` method of a `~numpy.random.Generator` instance instead; please see the :ref:`random-quick-start`. Parameters ---------- p : float or array_like of floats The probability of success of an individual trial. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``p`` is a scalar. Otherwise, ``np.array(p).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized geometric distribution. See Also -------- random.Generator.geometric: which should be used for new code. Examples -------- Draw ten thousand values from the geometric distribution, with the probability of an individual success equal to 0.35: >>> z = np.random.geometric(p=0.35, size=10000) How many trials succeeded after a single run? >>> (z == 1).sum() / 10000. 0.34889999999999999 #random gamma(shape, scale=1.0, size=None) Draw samples from a Gamma distribution. Samples are drawn from a Gamma distribution with specified parameters, `shape` (sometimes designated "k") and `scale` (sometimes designated "theta"), where both parameters are > 0. .. note:: New code should use the `~numpy.random.Generator.gamma` method of a `~numpy.random.Generator` instance instead; please see the :ref:`random-quick-start`. Parameters ---------- shape : float or array_like of floats The shape of the gamma distribution. Must be non-negative. scale : float or array_like of floats, optional The scale of the gamma distribution. Must be non-negative. Default is equal to 1. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``shape`` and ``scale`` are both scalars. Otherwise, ``np.broadcast(shape, scale).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized gamma distribution. See Also -------- scipy.stats.gamma : probability density function, distribution or cumulative density function, etc. random.Generator.gamma: which should be used for new code. Notes ----- The probability density for the Gamma distribution is .. math:: p(x) = x^{k-1}\frac{e^{-x/\theta}}{\theta^k\Gamma(k)}, where :math:`k` is the shape and :math:`\theta` the scale, and :math:`\Gamma` is the Gamma function. The Gamma distribution is often used to model the times to failure of electronic components, and arises naturally in processes for which the waiting times between Poisson distributed events are relevant. References ---------- .. [1] Weisstein, Eric W. "Gamma Distribution." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/GammaDistribution.html .. [2] Wikipedia, "Gamma distribution", https://en.wikipedia.org/wiki/Gamma_distribution Examples -------- Draw samples from the distribution: >>> shape, scale = 2., 2. # mean=4, std=2*sqrt(2) >>> s = np.random.gamma(shape, scale, 1000) Display the histogram of the samples, along with the probability density function: >>> import matplotlib.pyplot as plt >>> import scipy.special as sps # doctest: +SKIP >>> count, bins, ignored = plt.hist(s, 50, density=True) >>> y = bins**(shape-1)*(np.exp(-bins/scale) / # doctest: +SKIP ... (sps.gamma(shape)*scale**shape)) >>> plt.plot(bins, y, linewidth=2, color='r') # doctest: +SKIP >>> plt.show() f(dfnum, dfden, size=None) Draw samples from an F distribution. Samples are drawn from an F distribution with specified parameters, `dfnum` (degrees of freedom in numerator) and `dfden` (degrees of freedom in denominator), where both parameters must be greater than zero. The random variate of the F distribution (also known as the Fisher distribution) is a continuous probability distribution that arises in ANOVA tests, and is the ratio of two chi-square variates. .. note:: New code should use the `~numpy.random.Generator.f` method of a `~numpy.random.Generator` instance instead; please see the :ref:`random-quick-start`. Parameters ---------- dfnum : float or array_like of floats Degrees of freedom in numerator, must be > 0. dfden : float or array_like of float Degrees of freedom in denominator, must be > 0. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``dfnum`` and ``dfden`` are both scalars. Otherwise, ``np.broadcast(dfnum, dfden).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized Fisher distribution. See Also -------- scipy.stats.f : probability density function, distribution or cumulative density function, etc. random.Generator.f: which should be used for new code. Notes ----- The F statistic is used to compare in-group variances to between-group variances. Calculating the distribution depends on the sampling, and so it is a function of the respective degrees of freedom in the problem. The variable `dfnum` is the number of samples minus one, the between-groups degrees of freedom, while `dfden` is the within-groups degrees of freedom, the sum of the number of samples in each group minus the number of groups. References ---------- .. [1] Glantz, Stanton A. "Primer of Biostatistics.", McGraw-Hill, Fifth Edition, 2002. .. [2] Wikipedia, "F-distribution", https://en.wikipedia.org/wiki/F-distribution Examples -------- An example from Glantz[1], pp 47-40: Two groups, children of diabetics (25 people) and children from people without diabetes (25 controls). Fasting blood glucose was measured, case group had a mean value of 86.1, controls had a mean value of 82.2. Standard deviations were 2.09 and 2.49 respectively. Are these data consistent with the null hypothesis that the parents diabetic status does not affect their children's blood glucose levels? Calculating the F statistic from the data gives a value of 36.01. Draw samples from the distribution: >>> dfnum = 1. # between group degrees of freedom >>> dfden = 48. # within groups degrees of freedom >>> s = np.random.f(dfnum, dfden, 1000) The lower bound for the top 1% of the samples is : >>> np.sort(s)[-10] 7.61988120985 # random So there is about a 1% chance that the F statistic will exceed 7.62, the measured value is 36, so the null hypothesis is rejected at the 1% level. exponential(scale=1.0, size=None) Draw samples from an exponential distribution. Its probability density function is .. math:: f(x; \frac{1}{\beta}) = \frac{1}{\beta} \exp(-\frac{x}{\beta}), for ``x > 0`` and 0 elsewhere. :math:`\beta` is the scale parameter, which is the inverse of the rate parameter :math:`\lambda = 1/\beta`. The rate parameter is an alternative, widely used parameterization of the exponential distribution [3]_. The exponential distribution is a continuous analogue of the geometric distribution. It describes many common situations, such as the size of raindrops measured over many rainstorms [1]_, or the time between page requests to Wikipedia [2]_. .. note:: New code should use the `~numpy.random.Generator.exponential` method of a `~numpy.random.Generator` instance instead; please see the :ref:`random-quick-start`. Parameters ---------- scale : float or array_like of floats The scale parameter, :math:`\beta = 1/\lambda`. Must be non-negative. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``scale`` is a scalar. Otherwise, ``np.array(scale).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized exponential distribution. Examples -------- A real world example: Assume a company has 10000 customer support agents and the average time between customer calls is 4 minutes. >>> n = 10000 >>> time_between_calls = np.random.default_rng().exponential(scale=4, size=n) What is the probability that a customer will call in the next 4 to 5 minutes? >>> x = ((time_between_calls < 5).sum())/n >>> y = ((time_between_calls < 4).sum())/n >>> x-y 0.08 # may vary See Also -------- random.Generator.exponential: which should be used for new code. References ---------- .. [1] Peyton Z. Peebles Jr., "Probability, Random Variables and Random Signal Principles", 4th ed, 2001, p. 57. .. [2] Wikipedia, "Poisson process", https://en.wikipedia.org/wiki/Poisson_process .. [3] Wikipedia, "Exponential distribution", https://en.wikipedia.org/wiki/Exponential_distribution choice(a, size=None, replace=True, p=None) Generates a random sample from a given 1-D array .. versionadded:: 1.7.0 .. note:: New code should use the `~numpy.random.Generator.choice` method of a `~numpy.random.Generator` instance instead; please see the :ref:`random-quick-start`. Parameters ---------- a : 1-D array-like or int If an ndarray, a random sample is generated from its elements. If an int, the random sample is generated as if it were ``np.arange(a)`` size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. Default is None, in which case a single value is returned. replace : boolean, optional Whether the sample is with or without replacement. Default is True, meaning that a value of ``a`` can be selected multiple times. p : 1-D array-like, optional The probabilities associated with each entry in a. If not given, the sample assumes a uniform distribution over all entries in ``a``. Returns ------- samples : single item or ndarray The generated random samples Raises ------ ValueError If a is an int and less than zero, if a or p are not 1-dimensional, if a is an array-like of size 0, if p is not a vector of probabilities, if a and p have different lengths, or if replace=False and the sample size is greater than the population size See Also -------- randint, shuffle, permutation random.Generator.choice: which should be used in new code Notes ----- Setting user-specified probabilities through ``p`` uses a more general but less efficient sampler than the default. The general sampler produces a different sample than the optimized sampler even if each element of ``p`` is 1 / len(a). Sampling random rows from a 2-D array is not possible with this function, but is possible with `Generator.choice` through its ``axis`` keyword. Examples -------- Generate a uniform random sample from np.arange(5) of size 3: >>> np.random.choice(5, 3) array([0, 3, 4]) # random >>> #This is equivalent to np.random.randint(0,5,3) Generate a non-uniform random sample from np.arange(5) of size 3: >>> np.random.choice(5, 3, p=[0.1, 0, 0.3, 0.6, 0]) array([3, 3, 0]) # random Generate a uniform random sample from np.arange(5) of size 3 without replacement: >>> np.random.choice(5, 3, replace=False) array([3,1,0]) # random >>> #This is equivalent to np.random.permutation(np.arange(5))[:3] Generate a non-uniform random sample from np.arange(5) of size 3 without replacement: >>> np.random.choice(5, 3, replace=False, p=[0.1, 0, 0.3, 0.6, 0]) array([2, 3, 0]) # random Any of the above can be repeated with an arbitrary array-like instead of just integers. For instance: >>> aa_milne_arr = ['pooh', 'rabbit', 'piglet', 'Christopher'] >>> np.random.choice(aa_milne_arr, 5, p=[0.5, 0.1, 0.1, 0.3]) array(['pooh', 'pooh', 'pooh', 'Christopher', 'piglet'], # random dtype=' 0. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``df`` is a scalar. Otherwise, ``np.array(df).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized chi-square distribution. Raises ------ ValueError When `df` <= 0 or when an inappropriate `size` (e.g. ``size=-1``) is given. See Also -------- random.Generator.chisquare: which should be used for new code. Notes ----- The variable obtained by summing the squares of `df` independent, standard normally distributed random variables: .. math:: Q = \sum_{i=0}^{\mathtt{df}} X^2_i is chi-square distributed, denoted .. math:: Q \sim \chi^2_k. The probability density function of the chi-squared distribution is .. math:: p(x) = \frac{(1/2)^{k/2}}{\Gamma(k/2)} x^{k/2 - 1} e^{-x/2}, where :math:`\Gamma` is the gamma function, .. math:: \Gamma(x) = \int_0^{-\infty} t^{x - 1} e^{-t} dt. References ---------- .. [1] NIST "Engineering Statistics Handbook" https://www.itl.nist.gov/div898/handbook/eda/section3/eda3666.htm Examples -------- >>> np.random.chisquare(2,4) array([ 1.89920014, 9.00867716, 3.13710533, 5.62318272]) # random bytes(length) Return random bytes. .. note:: New code should use the `~numpy.random.Generator.bytes` method of a `~numpy.random.Generator` instance instead; please see the :ref:`random-quick-start`. Parameters ---------- length : int Number of random bytes. Returns ------- out : bytes String of length `length`. See Also -------- random.Generator.bytes: which should be used for new code. Examples -------- >>> np.random.bytes(10) b' eh\x85\x022SZ\xbf\xa4' #random binomial(n, p, size=None) Draw samples from a binomial distribution. Samples are drawn from a binomial distribution with specified parameters, n trials and p probability of success where n an integer >= 0 and p is in the interval [0,1]. (n may be input as a float, but it is truncated to an integer in use) .. note:: New code should use the `~numpy.random.Generator.binomial` method of a `~numpy.random.Generator` instance instead; please see the :ref:`random-quick-start`. Parameters ---------- n : int or array_like of ints Parameter of the distribution, >= 0. Floats are also accepted, but they will be truncated to integers. p : float or array_like of floats Parameter of the distribution, >= 0 and <=1. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``n`` and ``p`` are both scalars. Otherwise, ``np.broadcast(n, p).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized binomial distribution, where each sample is equal to the number of successes over the n trials. See Also -------- scipy.stats.binom : probability density function, distribution or cumulative density function, etc. random.Generator.binomial: which should be used for new code. Notes ----- The probability density for the binomial distribution is .. math:: P(N) = \binom{n}{N}p^N(1-p)^{n-N}, where :math:`n` is the number of trials, :math:`p` is the probability of success, and :math:`N` is the number of successes. When estimating the standard error of a proportion in a population by using a random sample, the normal distribution works well unless the product p*n <=5, where p = population proportion estimate, and n = number of samples, in which case the binomial distribution is used instead. For example, a sample of 15 people shows 4 who are left handed, and 11 who are right handed. Then p = 4/15 = 27%. 0.27*15 = 4, so the binomial distribution should be used in this case. References ---------- .. [1] Dalgaard, Peter, "Introductory Statistics with R", Springer-Verlag, 2002. .. [2] Glantz, Stanton A. "Primer of Biostatistics.", McGraw-Hill, Fifth Edition, 2002. .. [3] Lentner, Marvin, "Elementary Applied Statistics", Bogden and Quigley, 1972. .. [4] Weisstein, Eric W. "Binomial Distribution." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/BinomialDistribution.html .. [5] Wikipedia, "Binomial distribution", https://en.wikipedia.org/wiki/Binomial_distribution Examples -------- Draw samples from the distribution: >>> n, p = 10, .5 # number of trials, probability of each trial >>> s = np.random.binomial(n, p, 1000) # result of flipping a coin 10 times, tested 1000 times. A real world example. A company drills 9 wild-cat oil exploration wells, each with an estimated probability of success of 0.1. All nine wells fail. What is the probability of that happening? Let's do 20,000 trials of the model, and count the number that generate zero positive results. >>> sum(np.random.binomial(9, 0.1, 20000) == 0)/20000. # answer = 0.38885, or 38%. a must be greater than 0 unless no samples are taken'a' cannot be empty unless no samples are takenUnsupported dtype %r for randintRandomState.standard_gamma (line 1562)RandomState.multivariate_normal (line 4057)RandomState.logseries (line 3968)RandomState.lognormal (line 2973)RandomState.hypergeometric (line 3833)RandomState.geometric (line 3771)RandomState.dirichlet (line 4393)RandomState.chisquare (line 1909) wald(mean, scale, size=None) Draw samples from a Wald, or inverse Gaussian, distribution. As the scale approaches infinity, the distribution becomes more like a Gaussian. Some references claim that the Wald is an inverse Gaussian with mean equal to 1, but this is by no means universal. The inverse Gaussian distribution was first studied in relationship to Brownian motion. In 1956 M.C.K. Tweedie used the name inverse Gaussian because there is an inverse relationship between the time to cover a unit distance and distance covered in unit time. .. note:: New code should use the `~numpy.random.Generator.wald` method of a `~numpy.random.Generator` instance instead; please see the :ref:`random-quick-start`. Parameters ---------- mean : float or array_like of floats Distribution mean, must be > 0. scale : float or array_like of floats Scale parameter, must be > 0. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``mean`` and ``scale`` are both scalars. Otherwise, ``np.broadcast(mean, scale).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized Wald distribution. See Also -------- random.Generator.wald: which should be used for new code. Notes ----- The probability density function for the Wald distribution is .. math:: P(x;mean,scale) = \sqrt{\frac{scale}{2\pi x^3}}e^ \frac{-scale(x-mean)^2}{2\cdotp mean^2x} As noted above the inverse Gaussian distribution first arise from attempts to model Brownian motion. It is also a competitor to the Weibull for use in reliability modeling and modeling stock returns and interest rate processes. References ---------- .. [1] Brighton Webs Ltd., Wald Distribution, https://web.archive.org/web/20090423014010/http://www.brighton-webs.co.uk:80/distributions/wald.asp .. [2] Chhikara, Raj S., and Folks, J. Leroy, "The Inverse Gaussian Distribution: Theory : Methodology, and Applications", CRC Press, 1988. .. [3] Wikipedia, "Inverse Gaussian distribution" https://en.wikipedia.org/wiki/Inverse_Gaussian_distribution Examples -------- Draw values from the distribution and plot the histogram: >>> import matplotlib.pyplot as plt >>> h = plt.hist(np.random.wald(3, 2, 100000), bins=200, density=True) >>> plt.show() standard_normal(size=None) Draw samples from a standard Normal distribution (mean=0, stdev=1). .. note:: New code should use the `~numpy.random.Generator.standard_normal` method of a `~numpy.random.Generator` instance instead; please see the :ref:`random-quick-start`. Parameters ---------- size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. Default is None, in which case a single value is returned. Returns ------- out : float or ndarray A floating-point array of shape ``size`` of drawn samples, or a single sample if ``size`` was not specified. See Also -------- normal : Equivalent function with additional ``loc`` and ``scale`` arguments for setting the mean and standard deviation. random.Generator.standard_normal: which should be used for new code. Notes ----- For random samples from the normal distribution with mean ``mu`` and standard deviation ``sigma``, use one of:: mu + sigma * np.random.standard_normal(size=...) np.random.normal(mu, sigma, size=...) Examples -------- >>> np.random.standard_normal() 2.1923875335537315 #random >>> s = np.random.standard_normal(8000) >>> s array([ 0.6888893 , 0.78096262, -0.89086505, ..., 0.49876311, # random -0.38672696, -0.4685006 ]) # random >>> s.shape (8000,) >>> s = np.random.standard_normal(size=(3, 4, 2)) >>> s.shape (3, 4, 2) Two-by-four array of samples from the normal distribution with mean 3 and standard deviation 2.5: >>> 3 + 2.5 * np.random.standard_normal(size=(2, 4)) array([[-4.49401501, 4.00950034, -1.81814867, 7.29718677], # random [ 0.39924804, 4.68456316, 4.99394529, 4.84057254]]) # random standard_gamma(shape, size=None) Draw samples from a standard Gamma distribution. Samples are drawn from a Gamma distribution with specified parameters, shape (sometimes designated "k") and scale=1. .. note:: New code should use the `~numpy.random.Generator.standard_gamma` method of a `~numpy.random.Generator` instance instead; please see the :ref:`random-quick-start`. Parameters ---------- shape : float or array_like of floats Parameter, must be non-negative. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``shape`` is a scalar. Otherwise, ``np.array(shape).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized standard gamma distribution. See Also -------- scipy.stats.gamma : probability density function, distribution or cumulative density function, etc. random.Generator.standard_gamma: which should be used for new code. Notes ----- The probability density for the Gamma distribution is .. math:: p(x) = x^{k-1}\frac{e^{-x/\theta}}{\theta^k\Gamma(k)}, where :math:`k` is the shape and :math:`\theta` the scale, and :math:`\Gamma` is the Gamma function. The Gamma distribution is often used to model the times to failure of electronic components, and arises naturally in processes for which the waiting times between Poisson distributed events are relevant. References ---------- .. [1] Weisstein, Eric W. "Gamma Distribution." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/GammaDistribution.html .. [2] Wikipedia, "Gamma distribution", https://en.wikipedia.org/wiki/Gamma_distribution Examples -------- Draw samples from the distribution: >>> shape, scale = 2., 1. # mean and width >>> s = np.random.standard_gamma(shape, 1000000) Display the histogram of the samples, along with the probability density function: >>> import matplotlib.pyplot as plt >>> import scipy.special as sps # doctest: +SKIP >>> count, bins, ignored = plt.hist(s, 50, density=True) >>> y = bins**(shape-1) * ((np.exp(-bins/scale))/ # doctest: +SKIP ... (sps.gamma(shape) * scale**shape)) >>> plt.plot(bins, y, linewidth=2, color='r') # doctest: +SKIP >>> plt.show() standard_exponential(size=None) Draw samples from the standard exponential distribution. `standard_exponential` is identical to the exponential distribution with a scale parameter of 1. .. note:: New code should use the `~numpy.random.Generator.standard_exponential` method of a `~numpy.random.Generator` instance instead; please see the :ref:`random-quick-start`. Parameters ---------- size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. Default is None, in which case a single value is returned. Returns ------- out : float or ndarray Drawn samples. See Also -------- random.Generator.standard_exponential: which should be used for new code. Examples -------- Output a 3x8000 array: >>> n = np.random.standard_exponential((3, 8000)) standard_cauchy(size=None) Draw samples from a standard Cauchy distribution with mode = 0. Also known as the Lorentz distribution. .. note:: New code should use the `~numpy.random.Generator.standard_cauchy` method of a `~numpy.random.Generator` instance instead; please see the :ref:`random-quick-start`. Parameters ---------- size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. Default is None, in which case a single value is returned. Returns ------- samples : ndarray or scalar The drawn samples. See Also -------- random.Generator.standard_cauchy: which should be used for new code. Notes ----- The probability density function for the full Cauchy distribution is .. math:: P(x; x_0, \gamma) = \frac{1}{\pi \gamma \bigl[ 1+ (\frac{x-x_0}{\gamma})^2 \bigr] } and the Standard Cauchy distribution just sets :math:`x_0=0` and :math:`\gamma=1` The Cauchy distribution arises in the solution to the driven harmonic oscillator problem, and also describes spectral line broadening. It also describes the distribution of values at which a line tilted at a random angle will cut the x axis. When studying hypothesis tests that assume normality, seeing how the tests perform on data from a Cauchy distribution is a good indicator of their sensitivity to a heavy-tailed distribution, since the Cauchy looks very much like a Gaussian distribution, but with heavier tails. References ---------- .. [1] NIST/SEMATECH e-Handbook of Statistical Methods, "Cauchy Distribution", https://www.itl.nist.gov/div898/handbook/eda/section3/eda3663.htm .. [2] Weisstein, Eric W. "Cauchy Distribution." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/CauchyDistribution.html .. [3] Wikipedia, "Cauchy distribution" https://en.wikipedia.org/wiki/Cauchy_distribution Examples -------- Draw samples and plot the distribution: >>> import matplotlib.pyplot as plt >>> s = np.random.standard_cauchy(1000000) >>> s = s[(s>-25) & (s<25)] # truncate distribution so it plots well >>> plt.hist(s, bins=100) >>> plt.show() shuffle(x) Modify a sequence in-place by shuffling its contents. This function only shuffles the array along the first axis of a multi-dimensional array. The order of sub-arrays is changed but their contents remains the same. .. note:: New code should use the `~numpy.random.Generator.shuffle` method of a `~numpy.random.Generator` instance instead; please see the :ref:`random-quick-start`. Parameters ---------- x : ndarray or MutableSequence The array, list or mutable sequence to be shuffled. Returns ------- None See Also -------- random.Generator.shuffle: which should be used for new code. Examples -------- >>> arr = np.arange(10) >>> np.random.shuffle(arr) >>> arr [1 7 5 2 9 4 3 6 0 8] # random Multi-dimensional arrays are only shuffled along the first axis: >>> arr = np.arange(9).reshape((3, 3)) >>> np.random.shuffle(arr) >>> arr array([[3, 4, 5], # random [6, 7, 8], [0, 1, 2]]) seed(seed=None) Reseed a legacy MT19937 BitGenerator Notes ----- This is a convenience, legacy function. The best practice is to **not** reseed a BitGenerator, rather to recreate a new one. This method is here for legacy reasons. This example demonstrates best practice. >>> from numpy.random import MT19937 >>> from numpy.random import RandomState, SeedSequence >>> rs = RandomState(MT19937(SeedSequence(123456789))) # Later, you want to restart the stream >>> rs = RandomState(MT19937(SeedSequence(987654321))) random_sample(size=None) Return random floats in the half-open interval [0.0, 1.0). Results are from the "continuous uniform" distribution over the stated interval. To sample :math:`Unif[a, b), b > a` multiply the output of `random_sample` by `(b-a)` and add `a`:: (b - a) * random_sample() + a .. note:: New code should use the `~numpy.random.Generator.random` method of a `~numpy.random.Generator` instance instead; please see the :ref:`random-quick-start`. Parameters ---------- size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. Default is None, in which case a single value is returned. Returns ------- out : float or ndarray of floats Array of random floats of shape `size` (unless ``size=None``, in which case a single float is returned). See Also -------- random.Generator.random: which should be used for new code. Examples -------- >>> np.random.random_sample() 0.47108547995356098 # random >>> type(np.random.random_sample()) >>> np.random.random_sample((5,)) array([ 0.30220482, 0.86820401, 0.1654503 , 0.11659149, 0.54323428]) # random Three-by-two array of random numbers from [-5, 0): >>> 5 * np.random.random_sample((3, 2)) - 5 array([[-3.99149989, -0.52338984], # random [-2.99091858, -0.79479508], [-1.23204345, -1.75224494]]) randn(d0, d1, ..., dn) Return a sample (or samples) from the "standard normal" distribution. .. note:: This is a convenience function for users porting code from Matlab, and wraps `standard_normal`. That function takes a tuple to specify the size of the output, which is consistent with other NumPy functions like `numpy.zeros` and `numpy.ones`. .. note:: New code should use the `~numpy.random.Generator.standard_normal` method of a `~numpy.random.Generator` instance instead; please see the :ref:`random-quick-start`. If positive int_like arguments are provided, `randn` generates an array of shape ``(d0, d1, ..., dn)``, filled with random floats sampled from a univariate "normal" (Gaussian) distribution of mean 0 and variance 1. A single float randomly sampled from the distribution is returned if no argument is provided. Parameters ---------- d0, d1, ..., dn : int, optional The dimensions of the returned array, must be non-negative. If no argument is given a single Python float is returned. Returns ------- Z : ndarray or float A ``(d0, d1, ..., dn)``-shaped array of floating-point samples from the standard normal distribution, or a single such float if no parameters were supplied. See Also -------- standard_normal : Similar, but takes a tuple as its argument. normal : Also accepts mu and sigma arguments. random.Generator.standard_normal: which should be used for new code. Notes ----- For random samples from the normal distribution with mean ``mu`` and standard deviation ``sigma``, use:: sigma * np.random.randn(...) + mu Examples -------- >>> np.random.randn() 2.1923875335537315 # random Two-by-four array of samples from the normal distribution with mean 3 and standard deviation 2.5: >>> 3 + 2.5 * np.random.randn(2, 4) array([[-4.49401501, 4.00950034, -1.81814867, 7.29718677], # random [ 0.39924804, 4.68456316, 4.99394529, 4.84057254]]) # random rand(d0, d1, ..., dn) Random values in a given shape. .. note:: This is a convenience function for users porting code from Matlab, and wraps `random_sample`. That function takes a tuple to specify the size of the output, which is consistent with other NumPy functions like `numpy.zeros` and `numpy.ones`. Create an array of the given shape and populate it with random samples from a uniform distribution over ``[0, 1)``. Parameters ---------- d0, d1, ..., dn : int, optional The dimensions of the returned array, must be non-negative. If no argument is given a single Python float is returned. Returns ------- out : ndarray, shape ``(d0, d1, ..., dn)`` Random values. See Also -------- random Examples -------- >>> np.random.rand(3,2) array([[ 0.14022471, 0.96360618], #random [ 0.37601032, 0.25528411], #random [ 0.49313049, 0.94909878]]) #random poisson(lam=1.0, size=None) Draw samples from a Poisson distribution. The Poisson distribution is the limit of the binomial distribution for large N. .. note:: New code should use the `~numpy.random.Generator.poisson` method of a `~numpy.random.Generator` instance instead; please see the :ref:`random-quick-start`. Parameters ---------- lam : float or array_like of floats Expected number of events occurring in a fixed-time interval, must be >= 0. A sequence must be broadcastable over the requested size. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``lam`` is a scalar. Otherwise, ``np.array(lam).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized Poisson distribution. See Also -------- random.Generator.poisson: which should be used for new code. Notes ----- The Poisson distribution .. math:: f(k; \lambda)=\frac{\lambda^k e^{-\lambda}}{k!} For events with an expected separation :math:`\lambda` the Poisson distribution :math:`f(k; \lambda)` describes the probability of :math:`k` events occurring within the observed interval :math:`\lambda`. Because the output is limited to the range of the C int64 type, a ValueError is raised when `lam` is within 10 sigma of the maximum representable value. References ---------- .. [1] Weisstein, Eric W. "Poisson Distribution." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/PoissonDistribution.html .. [2] Wikipedia, "Poisson distribution", https://en.wikipedia.org/wiki/Poisson_distribution Examples -------- Draw samples from the distribution: >>> import numpy as np >>> s = np.random.poisson(5, 10000) Display histogram of the sample: >>> import matplotlib.pyplot as plt >>> count, bins, ignored = plt.hist(s, 14, density=True) >>> plt.show() Draw each 100 values for lambda 100 and 500: >>> s = np.random.poisson(lam=(100., 500.), size=(100, 2)) permutation(x) Randomly permute a sequence, or return a permuted range. If `x` is a multi-dimensional array, it is only shuffled along its first index. .. note:: New code should use the `~numpy.random.Generator.permutation` method of a `~numpy.random.Generator` instance instead; please see the :ref:`random-quick-start`. Parameters ---------- x : int or array_like If `x` is an integer, randomly permute ``np.arange(x)``. If `x` is an array, make a copy and shuffle the elements randomly. Returns ------- out : ndarray Permuted sequence or array range. See Also -------- random.Generator.permutation: which should be used for new code. Examples -------- >>> np.random.permutation(10) array([1, 7, 4, 3, 0, 9, 2, 5, 8, 6]) # random >>> np.random.permutation([1, 4, 9, 12, 15]) array([15, 1, 9, 4, 12]) # random >>> arr = np.arange(9).reshape((3, 3)) >>> np.random.permutation(arr) array([[6, 7, 8], # random [0, 1, 2], [3, 4, 5]]) ' object which is not a subclass of 'Sequence'; `shuffle` is not guaranteed to behave correctly. E.g., non-numpy array/tensor objects with view semantics may contain duplicates after shuffling. laplace(loc=0.0, scale=1.0, size=None) Draw samples from the Laplace or double exponential distribution with specified location (or mean) and scale (decay). The Laplace distribution is similar to the Gaussian/normal distribution, but is sharper at the peak and has fatter tails. It represents the difference between two independent, identically distributed exponential random variables. .. note:: New code should use the `~numpy.random.Generator.laplace` method of a `~numpy.random.Generator` instance instead; please see the :ref:`random-quick-start`. Parameters ---------- loc : float or array_like of floats, optional The position, :math:`\mu`, of the distribution peak. Default is 0. scale : float or array_like of floats, optional :math:`\lambda`, the exponential decay. Default is 1. Must be non- negative. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``loc`` and ``scale`` are both scalars. Otherwise, ``np.broadcast(loc, scale).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized Laplace distribution. See Also -------- random.Generator.laplace: which should be used for new code. Notes ----- It has the probability density function .. math:: f(x; \mu, \lambda) = \frac{1}{2\lambda} \exp\left(-\frac{|x - \mu|}{\lambda}\right). The first law of Laplace, from 1774, states that the frequency of an error can be expressed as an exponential function of the absolute magnitude of the error, which leads to the Laplace distribution. For many problems in economics and health sciences, this distribution seems to model the data better than the standard Gaussian distribution. References ---------- .. [1] Abramowitz, M. and Stegun, I. A. (Eds.). "Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing," New York: Dover, 1972. .. [2] Kotz, Samuel, et. al. "The Laplace Distribution and Generalizations, " Birkhauser, 2001. .. [3] Weisstein, Eric W. "Laplace Distribution." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/LaplaceDistribution.html .. [4] Wikipedia, "Laplace distribution", https://en.wikipedia.org/wiki/Laplace_distribution Examples -------- Draw samples from the distribution >>> loc, scale = 0., 1. >>> s = np.random.laplace(loc, scale, 1000) Display the histogram of the samples, along with the probability density function: >>> import matplotlib.pyplot as plt >>> count, bins, ignored = plt.hist(s, 30, density=True) >>> x = np.arange(-8., 8., .01) >>> pdf = np.exp(-abs(x-loc)/scale)/(2.*scale) >>> plt.plot(x, pdf) Plot Gaussian for comparison: >>> g = (1/(scale * np.sqrt(2 * np.pi)) * ... np.exp(-(x - loc)**2 / (2 * scale**2))) >>> plt.plot(x,g) dirichlet(alpha, size=None) Draw samples from the Dirichlet distribution. Draw `size` samples of dimension k from a Dirichlet distribution. A Dirichlet-distributed random variable can be seen as a multivariate generalization of a Beta distribution. The Dirichlet distribution is a conjugate prior of a multinomial distribution in Bayesian inference. .. note:: New code should use the `~numpy.random.Generator.dirichlet` method of a `~numpy.random.Generator` instance instead; please see the :ref:`random-quick-start`. Parameters ---------- alpha : sequence of floats, length k Parameter of the distribution (length ``k`` for sample of length ``k``). size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n)``, then ``m * n * k`` samples are drawn. Default is None, in which case a vector of length ``k`` is returned. Returns ------- samples : ndarray, The drawn samples, of shape ``(size, k)``. Raises ------ ValueError If any value in ``alpha`` is less than or equal to zero See Also -------- random.Generator.dirichlet: which should be used for new code. Notes ----- The Dirichlet distribution is a distribution over vectors :math:`x` that fulfil the conditions :math:`x_i>0` and :math:`\sum_{i=1}^k x_i = 1`. The probability density function :math:`p` of a Dirichlet-distributed random vector :math:`X` is proportional to .. math:: p(x) \propto \prod_{i=1}^{k}{x^{\alpha_i-1}_i}, where :math:`\alpha` is a vector containing the positive concentration parameters. The method uses the following property for computation: let :math:`Y` be a random vector which has components that follow a standard gamma distribution, then :math:`X = \frac{1}{\sum_{i=1}^k{Y_i}} Y` is Dirichlet-distributed References ---------- .. [1] David McKay, "Information Theory, Inference and Learning Algorithms," chapter 23, http://www.inference.org.uk/mackay/itila/ .. [2] Wikipedia, "Dirichlet distribution", https://en.wikipedia.org/wiki/Dirichlet_distribution Examples -------- Taking an example cited in Wikipedia, this distribution can be used if one wanted to cut strings (each of initial length 1.0) into K pieces with different lengths, where each piece had, on average, a designated average length, but allowing some variation in the relative sizes of the pieces. >>> s = np.random.dirichlet((10, 5, 3), 20).transpose() >>> import matplotlib.pyplot as plt >>> plt.barh(range(20), s[0]) >>> plt.barh(range(20), s[1], left=s[0], color='g') >>> plt.barh(range(20), s[2], left=s[0]+s[1], color='r') >>> plt.title("Lengths of Strings") RandomState.vonmises (line 2264)RandomState.rayleigh (line 3089)RandomState.logistic (line 2887)RandomState.binomial (line 3352)state dictionary is not valid.probabilities do not sum to 1RandomState.weibull (line 2456)RandomState.uniform (line 1049)RandomState.tomaxint (line 620)RandomState.shuffle (line 4542)RandomState.poisson (line 3592)RandomState.laplace (line 2669)pvals must be a 1-d sequenceRandomState.randint (line 678)RandomState.pareto (line 2353)RandomState.normal (line 1453)RandomState.gumbel (line 2763)'a' and 'p' must have same sizeRandomState.randn (line 1220)RandomState.power (line 2560)RandomState.gamma (line 1644)RandomState.choice (line 840)mean must be 1 dimensionalRange exceeds valid boundsRandomState.zipf (line 3675)RandomState.wald (line 3166)RandomState.rand (line 1176)RandomState.bytes (line 804)probabilities contain NaNRandomState.seed (line 228)'p' must be 1-dimensionalnumpy/random/mtrand.pyxa must be 1-dimensionalRandomState.f (line 1728)standard_exponentialnoncentral_chisquareyou are shuffling a 'numpy.random.mtrandmultivariate_normalngood + nbad < nsamplecline_in_tracebackarray is read-onlyDeprecationWarningset_bit_generatornegative_binomialget_bit_generator__randomstate_ctormay_share_memorysum(pvals[:-1]) > 1.0standard_normalstandard_cauchyrandom_integers_poisson_lam_maxcollections.abc_bit_generatorstandard_gamma_legacy_seedinghypergeometricRuntimeWarningrandom_samplecount_nonzero__class_getitem__bit_generatorOverflowErrorsearchsortedreturn_indexnumpy.linalgnoncentral_fnewbyteorder_initializingpermutationmultinomialexponentialcheck_validUserWarningRandomStateImportErrortriangularstandard_tstacklevel__pyx_vtable__mode > rightlogical_orless_equalleft == rightissubdtypeempty_likeValueErrorIndexErrorwriteablesingletonset_statelogserieslognormalleft > modeisenabledhas_gaussget_stategeometricdirichletchisquareTypeErrorMT19937warningsvonmisessubtractreversedreducerayleighoperatorlogisticitemsizeisscalarisnativeisfinitebinomialallcloseSequenceweibulluniformtobytesstridesshufflereshapereplacerandintpoissonnsample_mt19937laplacegreaterfloat64disablecastingcapsulebg_type at 0x{:X}asarrayalpha <= 0_MT19937unsafeuniqueuint64uint32uint16sample__reduce__random_rand_pickleparetoobject_normallengthlegacykwargs__import__ignoregumbelformatenabledoublecumsumchoicebitgenastypearangezerosuint8statesigmashapescalerightravelrangerandnraisepvalspowernumpyngoodkappaisnanint64int32int16indexgaussgammaflagsfinfoequal__enter__emptydtypedfnumdfden__class__bytesarrayalpha__all__zipfwarnwaldtype__test__takesqrt__spec__sortsizesideseedrtolranfrandprodnoncndimnbad__name__modemean__main__locklessleftitemintpint8high__exit__copybool_betaatolargstolsvdsum__str__poslowloclamkeygetepsdotcovanyalladd?*qh???/*p?3f?(_?xY?յS?N?J?F?DB?Qt??u+ ?A?Į?"?ʝ?G??i>l>7>>*J>>>^>>F>>7P>>K{>>>u>;->>>|>eO>4(>8>L>N>ȿ>>>>·>ε>߳>>>0>U>~>|>ަ>Y>IP>w>ҟ>B>e>)>~>_>>C>{>>J>﷍>X(>'>N>Í>x >b>x>>!>^}>;z>Хw>@t>wr>byl>i>g>Sd>3a>^>]\>&Y>z)W>T>P R>O>L>5~J>3H>E>nC>@>VK>>;>9>07>4>2>>0>p->+>s)>7'>%>"> >s>L>*>= >T>T>4>>y >ϣ > >>L>>l>=+==0 =C==8==hp==0== =n==|===;=ں=Z=o,=ް=ߗ=.W===%==r=W= C=4=M,=4*=D.=y8=H=~=x=3E3lM3FT3/[3b3i34p3fw3&~3[3B3ψ3g37!3>3T3d3n3r3Fq3j3_31P3r<3$3k 33ȸ3q3|{3P3#3C333dY3"3+3®3r35333x37333p3-33731b33l44(44h4C44 4` 4M 47 44?4nB44L4 i4a4T044542`44p. 4!4"4i$4%4@'4t(4>*4+4,4j.4/4'P1424):4454&)7484c:4;4$=4+>4@4A4KC4vD4B(F4G4:I4J4rTL4M4GuO4Q4R44T4U4EiW4Y4 Z4G\4]4_4:a4b4d4Bf4\g4ji4bk4m4n4p4\r4}"t4Yu4Hw4[y4X{46.}4 4q4a4]S4F4N<434,4+(4{%4$4o&4,*4'04m84 C4P4_4q474{4w4>ԕ44s4<4d444$4 (4a44lߢ4$4l44x 4_444{4 4EP4±4{:4귴4);4nķ4S444<*4տ44A4.44ע44f4RW4R4*Y4Fk44δ444444g44k4<444y44u4_45555@5ó 5 5]5^555q5v 5!5%5V*5s/5;S55:<5D5NO5^5Nv5QHqoMֻanjDotTrotou$w'xx,jyy7\z׻z{W{S{{.|3|]|ȃ|||I||}C0}F}Z}m}}S}(}}-}}"}}|}M}~i ~~~B(~o0~C8~?~F~M~T~Z~a~f~l~r~]w~v|~`~ ~~$~m~~~w~:~ަ~f~ѭ~#~Z~y~~q~K~~~^~~a~~~`~~~~~~~~f~*~~~-~~J~~=~~~\~~~$~U~}~~~~~pH`  i   6  H  A!B+m 5XttW3 `wK\ L   s   G {V~~~d~~x~~K~~~~~~~~~)~~~a~~~{~;~~A~~~m~~z~~~"~k~]~~~ԃ~|~s~j~Ua~W~K~?~2~$~~~ }}} }}i}A}}|Q|D|{3N{zeyww7ms?7E?P?'{{?*!?bv?mU?9U1T?/v?x]?&1$-?~ n?cK[!?I?\Omg?f?uLi=?sڂl?x?Qf?ij?%ᨯC?+?Dܻ?z?cE#;?^E#?$O?2m?P"K?>?{s?%;?omo?3;?J9?++?*T[?};1s?HeC?$`?vE!=?ſ-r?MBц?K=?Q}6Ei?7u? !?z}k? ~?@?`x?*?8? 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