Initialisation du repository de Beta

venv/lib/python3.12/site-packages/sympy/stats/__init__.py

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+"""
+SymPy statistics module
+
+Introduces a random variable type into the SymPy language.
+
+Random variables may be declared using prebuilt functions such as
+Normal, Exponential, Coin, Die, etc...  or built with functions like FiniteRV.
+
+Queries on random expressions can be made using the functions
+
+========================= =============================
+    Expression                    Meaning
+------------------------- -----------------------------
+ ``P(condition)``          Probability
+ ``E(expression)``         Expected value
+ ``H(expression)``         Entropy
+ ``variance(expression)``  Variance
+ ``density(expression)``   Probability Density Function
+ ``sample(expression)``    Produce a realization
+ ``where(condition)``      Where the condition is true
+========================= =============================
+
+Examples
+========
+
+>>> from sympy.stats import P, E, variance, Die, Normal
+>>> from sympy import simplify
+>>> X, Y = Die('X', 6), Die('Y', 6) # Define two six sided dice
+>>> Z = Normal('Z', 0, 1) # Declare a Normal random variable with mean 0, std 1
+>>> P(X>3) # Probability X is greater than 3
+1/2
+>>> E(X+Y) # Expectation of the sum of two dice
+7
+>>> variance(X+Y) # Variance of the sum of two dice
+35/6
+>>> simplify(P(Z>1)) # Probability of Z being greater than 1
+1/2 - erf(sqrt(2)/2)/2
+
+
+One could also create custom distribution and define custom random variables
+as follows:
+
+1. If you want to create a Continuous Random Variable:
+
+>>> from sympy.stats import ContinuousRV, P, E
+>>> from sympy import exp, Symbol, Interval, oo
+>>> x = Symbol('x')
+>>> pdf = exp(-x) # pdf of the Continuous Distribution
+>>> Z = ContinuousRV(x, pdf, set=Interval(0, oo))
+>>> E(Z)
+1
+>>> P(Z > 5)
+exp(-5)
+
+1.1 To create an instance of Continuous Distribution:
+
+>>> from sympy.stats import ContinuousDistributionHandmade
+>>> from sympy import Lambda
+>>> dist = ContinuousDistributionHandmade(Lambda(x, pdf), set=Interval(0, oo))
+>>> dist.pdf(x)
+exp(-x)
+
+2. If you want to create a Discrete Random Variable:
+
+>>> from sympy.stats import DiscreteRV, P, E
+>>> from sympy import Symbol, S
+>>> p = S(1)/2
+>>> x = Symbol('x', integer=True, positive=True)
+>>> pdf = p*(1 - p)**(x - 1)
+>>> D = DiscreteRV(x, pdf, set=S.Naturals)
+>>> E(D)
+2
+>>> P(D > 3)
+1/8
+
+2.1 To create an instance of Discrete Distribution:
+
+>>> from sympy.stats import DiscreteDistributionHandmade
+>>> from sympy import Lambda
+>>> dist = DiscreteDistributionHandmade(Lambda(x, pdf), set=S.Naturals)
+>>> dist.pdf(x)
+2**(1 - x)/2
+
+3. If you want to create a Finite Random Variable:
+
+>>> from sympy.stats import FiniteRV, P, E
+>>> from sympy import Rational, Eq
+>>> pmf = {1: Rational(1, 3), 2: Rational(1, 6), 3: Rational(1, 4), 4: Rational(1, 4)}
+>>> X = FiniteRV('X', pmf)
+>>> E(X)
+29/12
+>>> P(X > 3)
+1/4
+
+3.1 To create an instance of Finite Distribution:
+
+>>> from sympy.stats import FiniteDistributionHandmade
+>>> dist = FiniteDistributionHandmade(pmf)
+>>> dist.pmf(x)
+Lambda(x, Piecewise((1/3, Eq(x, 1)), (1/6, Eq(x, 2)), (1/4, Eq(x, 3) | Eq(x, 4)), (0, True)))
+"""
+
+__all__ = [
+    'P', 'E', 'H', 'density', 'where', 'given', 'sample', 'cdf','median',
+    'characteristic_function', 'pspace', 'sample_iter', 'variance', 'std',
+    'skewness', 'kurtosis', 'covariance', 'dependent', 'entropy', 'independent',
+    'random_symbols', 'correlation', 'factorial_moment', 'moment', 'cmoment',
+    'sampling_density', 'moment_generating_function', 'smoment', 'quantile',
+    'coskewness', 'sample_stochastic_process',
+
+    'FiniteRV', 'DiscreteUniform', 'Die', 'Bernoulli', 'Coin', 'Binomial',
+    'BetaBinomial', 'Hypergeometric', 'Rademacher', 'IdealSoliton', 'RobustSoliton',
+    'FiniteDistributionHandmade',
+
+    'ContinuousRV', 'Arcsin', 'Benini', 'Beta', 'BetaNoncentral', 'BetaPrime',
+    'BoundedPareto', 'Cauchy', 'Chi', 'ChiNoncentral', 'ChiSquared', 'Dagum', 'Davis', 'Erlang',
+    'ExGaussian', 'Exponential', 'ExponentialPower', 'FDistribution',
+    'FisherZ', 'Frechet', 'Gamma', 'GammaInverse', 'Gompertz', 'Gumbel',
+    'Kumaraswamy', 'Laplace', 'Levy', 'Logistic','LogCauchy', 'LogLogistic', 'LogitNormal', 'LogNormal', 'Lomax',
+    'Moyal', 'Maxwell', 'Nakagami', 'Normal', 'GaussianInverse', 'Pareto', 'PowerFunction',
+    'QuadraticU', 'RaisedCosine', 'Rayleigh','Reciprocal', 'StudentT', 'ShiftedGompertz',
+    'Trapezoidal', 'Triangular', 'Uniform', 'UniformSum', 'VonMises', 'Wald',
+    'Weibull', 'WignerSemicircle', 'ContinuousDistributionHandmade',
+
+    'FlorySchulz', 'Geometric','Hermite', 'Logarithmic', 'NegativeBinomial', 'Poisson', 'Skellam',
+    'YuleSimon', 'Zeta', 'DiscreteRV', 'DiscreteDistributionHandmade',
+
+    'JointRV', 'Dirichlet', 'GeneralizedMultivariateLogGamma',
+    'GeneralizedMultivariateLogGammaOmega', 'Multinomial', 'MultivariateBeta',
+    'MultivariateEwens', 'MultivariateT', 'NegativeMultinomial',
+    'NormalGamma', 'MultivariateNormal', 'MultivariateLaplace', 'marginal_distribution',
+
+    'StochasticProcess', 'DiscreteTimeStochasticProcess',
+    'DiscreteMarkovChain', 'TransitionMatrixOf', 'StochasticStateSpaceOf',
+    'GeneratorMatrixOf', 'ContinuousMarkovChain', 'BernoulliProcess',
+    'PoissonProcess', 'WienerProcess', 'GammaProcess',
+
+    'CircularEnsemble', 'CircularUnitaryEnsemble',
+    'CircularOrthogonalEnsemble', 'CircularSymplecticEnsemble',
+    'GaussianEnsemble', 'GaussianUnitaryEnsemble',
+    'GaussianOrthogonalEnsemble', 'GaussianSymplecticEnsemble',
+    'joint_eigen_distribution', 'JointEigenDistribution',
+    'level_spacing_distribution',
+
+    'MatrixGamma', 'Wishart', 'MatrixNormal', 'MatrixStudentT',
+
+    'Probability', 'Expectation', 'Variance', 'Covariance', 'Moment',
+    'CentralMoment',
+
+    'ExpectationMatrix', 'VarianceMatrix', 'CrossCovarianceMatrix'
+
+]
+from .rv_interface import (P, E, H, density, where, given, sample, cdf, median,
+        characteristic_function, pspace, sample_iter, variance, std, skewness,
+        kurtosis, covariance, dependent, entropy, independent, random_symbols,
+        correlation, factorial_moment, moment, cmoment, sampling_density,
+        moment_generating_function, smoment, quantile, coskewness,
+        sample_stochastic_process)
+
+from .frv_types import (FiniteRV, DiscreteUniform, Die, Bernoulli, Coin,
+        Binomial, BetaBinomial, Hypergeometric, Rademacher,
+        FiniteDistributionHandmade, IdealSoliton, RobustSoliton)
+
+from .crv_types import (ContinuousRV, Arcsin, Benini, Beta, BetaNoncentral,
+        BetaPrime, BoundedPareto, Cauchy, Chi, ChiNoncentral, ChiSquared,
+        Dagum, Davis, Erlang, ExGaussian, Exponential, ExponentialPower,
+        FDistribution, FisherZ, Frechet, Gamma, GammaInverse, GaussianInverse,
+        Gompertz, Gumbel, Kumaraswamy, Laplace, Levy, Logistic, LogCauchy,
+        LogLogistic, LogitNormal, LogNormal, Lomax, Maxwell, Moyal, Nakagami,
+        Normal, Pareto, QuadraticU, RaisedCosine, Rayleigh, Reciprocal,
+        StudentT, PowerFunction, ShiftedGompertz, Trapezoidal, Triangular,
+        Uniform, UniformSum, VonMises, Wald, Weibull, WignerSemicircle,
+        ContinuousDistributionHandmade)
+
+from .drv_types import (FlorySchulz, Geometric, Hermite, Logarithmic, NegativeBinomial, Poisson,
+        Skellam, YuleSimon, Zeta, DiscreteRV, DiscreteDistributionHandmade)
+
+from .joint_rv_types import (JointRV, Dirichlet,
+        GeneralizedMultivariateLogGamma, GeneralizedMultivariateLogGammaOmega,
+        Multinomial, MultivariateBeta, MultivariateEwens, MultivariateT,
+        NegativeMultinomial, NormalGamma, MultivariateNormal, MultivariateLaplace,
+        marginal_distribution)
+
+from .stochastic_process_types import (StochasticProcess,
+        DiscreteTimeStochasticProcess, DiscreteMarkovChain,
+        TransitionMatrixOf, StochasticStateSpaceOf, GeneratorMatrixOf,
+        ContinuousMarkovChain, BernoulliProcess, PoissonProcess, WienerProcess,
+        GammaProcess)
+
+from .random_matrix_models import (CircularEnsemble, CircularUnitaryEnsemble,
+        CircularOrthogonalEnsemble, CircularSymplecticEnsemble,
+        GaussianEnsemble, GaussianUnitaryEnsemble, GaussianOrthogonalEnsemble,
+        GaussianSymplecticEnsemble, joint_eigen_distribution,
+        JointEigenDistribution, level_spacing_distribution)
+
+from .matrix_distributions import MatrixGamma, Wishart, MatrixNormal, MatrixStudentT
+
+from .symbolic_probability import (Probability, Expectation, Variance,
+        Covariance, Moment, CentralMoment)
+
+from .symbolic_multivariate_probability import (ExpectationMatrix, VarianceMatrix,
+        CrossCovarianceMatrix)