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