K i'!dZgdZddlmZmZmZmZmZmZm Z m Z m Z m Z m Z mZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZm Z m!Z!ddl"m#Z#m$Z$m%Z%m&Z&m'Z'm(Z(m)Z)m*Z*m+Z+m,Z,m-Z-m.Z.ddl/m0Z0m1Z1m2Z2m3Z3m4Z4m5Z5m6Z6m7Z7m8Z8m9Z9m:Z:m;Z;mZ>m?Z?m@Z@mAZAmBZBmCZCmDZDmEZEmFZFmGZGmHZHmIZImJZJmKZKmLZLmMZMmNZNmOZOmPZPmQZQmRZRmSZSmTZTmUZUmVZVmWZWmXZXmYZYmZZZm[Z[m\Z\m]Z]m^Z^m_Z_m`Z`maZambZbmcZcmdZdmeZemfZfddlgmhZhmiZimjZjmkZkmlZlmmZmmnZnmoZompZpmqZqmrZrddlsmtZtmuZumvZvmwZwmxZxmyZymzZzm{Z{m|Z|m}Z}m~Z~mZmZddlmZmZmZmZmZmZmZmZmZmZmZdd lmZmZmZmZmZmZmZmZmZmZmZdd lmZmZmZmZdd lmZmZmZmZmZmZdd lmZmZmZy )a 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))) )PEHdensitywheregivensamplecdfmediancharacteristic_functionpspace sample_itervariancestdskewnesskurtosis covariance dependententropy independentrandom_symbols correlationfactorial_momentmomentcmomentsampling_densitymoment_generating_functionsmomentquantile coskewnesssample_stochastic_processFiniteRVDiscreteUniformDie BernoulliCoinBinomial BetaBinomialHypergeometric Rademacher IdealSoliton RobustSolitonFiniteDistributionHandmade ContinuousRVArcsinBeniniBetaBetaNoncentral BetaPrime BoundedParetoCauchyChi ChiNoncentral ChiSquaredDagumDavisErlang ExGaussian ExponentialExponentialPower FDistributionFisherZFrechetGamma GammaInverseGompertzGumbel KumaraswamyLaplaceLevyLogistic LogCauchy LogLogistic LogitNormal LogNormalLomaxMoyalMaxwellNakagamiNormalGaussianInversePareto PowerFunction QuadraticU RaisedCosineRayleigh ReciprocalStudentTShiftedGompertz Trapezoidal TriangularUniform UniformSumVonMisesWaldWeibullWignerSemicircleContinuousDistributionHandmade FlorySchulz GeometricHermite LogarithmicNegativeBinomialPoissonSkellam YuleSimonZeta DiscreteRVDiscreteDistributionHandmadeJointRV DirichletGeneralizedMultivariateLogGamma$GeneralizedMultivariateLogGammaOmega MultinomialMultivariateBetaMultivariateEwens MultivariateTNegativeMultinomial NormalGammaMultivariateNormalMultivariateLaplacemarginal_distributionStochasticProcessDiscreteTimeStochasticProcessDiscreteMarkovChainTransitionMatrixOfStochasticStateSpaceOfGeneratorMatrixOfContinuousMarkovChainBernoulliProcessPoissonProcess WienerProcess GammaProcessCircularEnsembleCircularUnitaryEnsembleCircularOrthogonalEnsembleCircularSymplecticEnsembleGaussianEnsembleGaussianUnitaryEnsembleGaussianOrthogonalEnsembleGaussianSymplecticEnsemblejoint_eigen_distributionJointEigenDistributionlevel_spacing_distribution MatrixGammaWishart MatrixNormalMatrixStudentT Probability ExpectationVariance CovarianceMoment CentralMomentExpectationMatrixVarianceMatrixCrossCovarianceMatrix)rrrrrrrr r r r r rrrrrrrrrrrrrrrrrrr ) r!r"r#r$r%r&r'r(r)r,r*r+)7r-r.r/r0r1r2r3r4r5r6r7r8r9r:r;r<r=r>r?r@rArBrRrCrDrErFrGrHrIrJrKrLrMrOrNrPrQrSrUrVrWrXrYrTrZr[r\r]r^r_r`rarbrc) rdrerfrgrhrirjrkrlrmrn) rorprqrrrsrtrurvrwrxryrzr{) r|r}r~rrrrrrrr) rrrrrrrrrrr)rrrr)rrrrrr)rrrN)__doc____all__ rv_interfacerrrrrrrr r r r r rrrrrrrrrrrrrrrrrrr frv_typesr!r"r#r$r%r&r'r(r)r,r*r+ crv_typesr-r.r/r0r1r2r3r4r5r6r7r8r9r:r;r<r=r>r?r@rArBrRrCrDrErFrGrHrIrJrKrLrMrOrNrPrQrSrUrVrWrXrYrTrZr[r\r]r^r_r`rarbrc drv_typesrdrerfrgrhrirjrkrlrmrnjoint_rv_typesrorprqrrrsrtrurvrwrxryrzr{stochastic_process_typesr|r}r~rrrrrrrrrandom_matrix_modelsrrrrrrrrrrrmatrix_distributionsrrrrsymbolic_probabilityrrrrrr!symbolic_multivariate_probabilityrrrZ/mnt/ssd/data/python-lab/Trading/venv/lib/python3.12/site-packages/sympy/stats/__init__.pyrsdL1 d#########AAAA ( ( ( ( ( ( ( ( ( ( ( ( ( ( (LLLL  <<<< UT++r