o 8VaY@sDddlZddlmZmZmZmZmZmZmZm Z m Z m Z ddl mZddlmZddlmZddlmZddlmZddlmZmZdd lmZmZmZmZmZm Z m!Z!m"Z"m#Z#m$Z$gd Z%e!&ed d Z'e!&ed d Z'GdddeZ(GdddeZ)GdddeZ*GdddeZ+GdddeZ,GdddeZ-dS)N) ExprAddMulSIntegralEqSumSymbolexpandNot)default_sort_key)global_parameters)_sympify) Relational)Boolean)variance covariance) RandomSymbolpspace dependentgiven sampling_ERandomIndexedSymbol is_randomPSpace sampling_Prandom_symbols) Probability ExpectationVariance CovariancecCs8|j}t|dkrtt||krdStdd|DS)NFcSsg|]}t|qS)r).0ir"r"B/usr/lib/python3/dist-packages/sympy/stats/symbolic_probability.py sz_..)Z free_symbolslennextiterany)xatomsr"r"r%_sr-cCsdS)NTr"r+r"r"r%r-sc@s8eZdZdZd ddZddZd ddZeZd d ZdS) ra Symbolic expression for the probability. Examples ======== >>> from sympy.stats import Probability, Normal >>> from sympy import Integral >>> X = Normal("X", 0, 1) >>> prob = Probability(X > 1) >>> prob Probability(X > 1) Integral representation: >>> prob.rewrite(Integral) Integral(sqrt(2)*exp(-_z**2/2)/(2*sqrt(pi)), (_z, 1, oo)) Evaluation of the integral: >>> prob.evaluate_integral() sqrt(2)*(-sqrt(2)*sqrt(pi)*erf(sqrt(2)/2) + sqrt(2)*sqrt(pi))/(4*sqrt(pi)) NcKs>t|}|durt||}n t|}t|||}||_|SN)rr__new__ _condition)clsZprob conditionkwargsobjr"r"r%r06szProbability.__new__c s|jd}|j|dd}|dd }t|tr.tj|j|jd|djd i|S| t r=t |j ||dStt r}t|}t|dkrg|dkrgddlm}|||jd i|ddStfdd |Drwt|St|Sdurtttfstd dks|tjurtjSt|ttfstd ||tjurtjS|rt||d Sdurtt|St |tkrt|St | |}t|d r|r|S|S)Nr numsamplesF for_rewriteevaluater!)BernoulliDistributioncsg|]}t|qSr")r)r#rvZgiven_conditionr"r%r&Sz$Probability.doit..z4%s is not a relational or combination of relationals)r6doitr")argsr1get isinstancer rOnefuncr>hasrrZ probabilityrrr'Zsympy.stats.frv_typesr:r*rrr ValueErrorZfalseZerotruerrrhasattr)selfhintsr3r6r7Zcondrvr:resultr"r<r%r>@s`            zProbability.doitcKs|j||djddS)Nr3T)r7rCr>rIargr3r4r"r"r%_eval_rewrite_as_Integraluz%Probability._eval_rewrite_as_IntegralcC|tSr/rewriterr>rIr"r"r%evaluate_integralzzProbability.evaluate_integralr/) __name__ __module__ __qualname____doc__r0r>rP_eval_rewrite_as_SumrVr"r"r"r%rs  5 rc@sNeZdZdZdddZddZddZdd d Zdd d ZeZ d dZ e Z dS)ra( Symbolic expression for the expectation. Examples ======== >>> from sympy.stats import Expectation, Normal, Probability, Poisson >>> from sympy import symbols, Integral, Sum >>> mu = symbols("mu") >>> sigma = symbols("sigma", positive=True) >>> X = Normal("X", mu, sigma) >>> Expectation(X) Expectation(X) >>> Expectation(X).evaluate_integral().simplify() mu To get the integral expression of the expectation: >>> Expectation(X).rewrite(Integral) Integral(sqrt(2)*X*exp(-(X - mu)**2/(2*sigma**2))/(2*sqrt(pi)*sigma), (X, -oo, oo)) The same integral expression, in more abstract terms: >>> Expectation(X).rewrite(Probability) Integral(x*Probability(Eq(X, x)), (x, -oo, oo)) To get the Summation expression of the expectation for discrete random variables: >>> lamda = symbols('lamda', positive=True) >>> Z = Poisson('Z', lamda) >>> Expectation(Z).rewrite(Sum) Sum(Z*lamda**Z*exp(-lamda)/factorial(Z), (Z, 0, oo)) This class is aware of some properties of the expectation: >>> from sympy.abc import a >>> Expectation(a*X) Expectation(a*X) >>> Y = Normal("Y", 1, 2) >>> Expectation(X + Y) Expectation(X + Y) To expand the ``Expectation`` into its expression, use ``expand()``: >>> Expectation(X + Y).expand() Expectation(X) + Expectation(Y) >>> Expectation(a*X + Y).expand() a*Expectation(X) + Expectation(Y) >>> Expectation(a*X + Y) Expectation(a*X + Y) >>> Expectation((X + Y)*(X - Y)).expand() Expectation(X**2) - Expectation(Y**2) To evaluate the ``Expectation``, use ``doit()``: >>> Expectation(X + Y).doit() mu + 1 >>> Expectation(X + Expectation(Y + Expectation(2*X))).doit() 3*mu + 1 To prevent evaluating nested ``Expectation``, use ``doit(deep=False)`` >>> Expectation(X + Expectation(Y)).doit(deep=False) mu + Expectation(Expectation(Y)) >>> Expectation(X + Expectation(Y + Expectation(2*X))).doit(deep=False) mu + Expectation(Expectation(Y + Expectation(2*X))) NcKsft|}|jrddlm}|||S|dur#t|s|St||}n t|}t|||}||_|S)Nr)ExpectationMatrix)r is_Matrix-sympy.stats.symbolic_multivariate_probabilityr]rrr0r1)r2exprr3r4r]r5r"r"r%r0s  zExpectation.__new__c s|jd}|jt|s|St|tr tfdd|jDSt|}t|tr6tfdd|jDSt|trbg}g}|jD]}t|rN||qB||qBt|t t|dS|S)Nrc3 |] }t|dVqdSrLNrr r#arLr"r% z%Expectation.expand..c3rarbrcrdrLr"r%rfrgrL) r?r1rrArfromiter_expandrappendr)rIrJr`Z expand_exprr;nonrvrer"rLr%r s,       zExpectation.expandc s@dd}jjd}dd}dd }|r%|jd i}t|r.t|tr0|S|r@dd}t|||dS|t rMt | |Sdur_ t |jd iS|jrptfd d |jDS|jrz|trz|St |tkr |St |j ||d }t|d r|r|jd iS|S)NdeepTrr6Fr7evalf)r6rmcs:g|]}t|ts|jdin|qS)r")rArrCr>)r#rOr3rJrIr"r%r& s  z$Expectation.doit..r8r>r")r@r1r?r>rrArrrDrrZcompute_expectationrCrZis_AddrZis_Mulr,rrH)rIrJrlr`r6r7rmrKr"rnr%r>s:       zExpectation.doitcKs|t}t|dkrtt|dkr|S|}|jdur#td|j}|jd r5t |j }nt |jd}|jj r\t |||tt|||||jjjj|jjjjfS|jjrbtt|||tt|||||jjjj|jjjfS)Nr!rzProbability space not knownZ_1)r,rr'NotImplementedErrorpoprrEsymbolnameisupperr lowerZ is_ContinuousrreplacerrdomainsetinfsupZ is_Finiter)rIrOr3r4Zrvsr;rqr"r"r%_eval_rewrite_as_Probabilitys"    86z(Expectation._eval_rewrite_as_ProbabilitycKs|j||djdddS)NrLFT)rlr7rMrNr"r"r%rP3sz%Expectation._eval_rewrite_as_IntegralcCrRr/rSrUr"r"r%rV8rWzExpectation.evaluate_integralr/) rXrYrZr[r0r r>rzrPr\rVZ evaluate_sumr"r"r"r%r~s E + rc@sLeZdZdZdddZddZdddZdd d Zdd d ZeZ d dZ dS)ra Symbolic expression for the variance. Examples ======== >>> from sympy import symbols, Integral >>> from sympy.stats import Normal, Expectation, Variance, Probability >>> mu = symbols("mu", positive=True) >>> sigma = symbols("sigma", positive=True) >>> X = Normal("X", mu, sigma) >>> Variance(X) Variance(X) >>> Variance(X).evaluate_integral() sigma**2 Integral representation of the underlying calculations: >>> Variance(X).rewrite(Integral) Integral(sqrt(2)*(X - Integral(sqrt(2)*X*exp(-(X - mu)**2/(2*sigma**2))/(2*sqrt(pi)*sigma), (X, -oo, oo)))**2*exp(-(X - mu)**2/(2*sigma**2))/(2*sqrt(pi)*sigma), (X, -oo, oo)) Integral representation, without expanding the PDF: >>> Variance(X).rewrite(Probability) -Integral(x*Probability(Eq(X, x)), (x, -oo, oo))**2 + Integral(x**2*Probability(Eq(X, x)), (x, -oo, oo)) Rewrite the variance in terms of the expectation >>> Variance(X).rewrite(Expectation) -Expectation(X)**2 + Expectation(X**2) Some transformations based on the properties of the variance may happen: >>> from sympy.abc import a >>> Y = Normal("Y", 0, 1) >>> Variance(a*X) Variance(a*X) To expand the variance in its expression, use ``expand()``: >>> Variance(a*X).expand() a**2*Variance(X) >>> Variance(X + Y) Variance(X + Y) >>> Variance(X + Y).expand() 2*Covariance(X, Y) + Variance(X) + Variance(Y) NcKsZt|}|jrddlm}|||S|durt||}n t|}t|||}||_|S)Nr)VarianceMatrix)rr^r_r{rr0r1)r2rOr3r4r{r5r"r"r%r0ns  zVariance.__new__c  s|jd}|jt|stjSt|tr|St|trLg}|jD] }t|r+||q tt fdd|}fdd}tt |t |d}||St|t rg}g}|jD]}t|rd||qX||dqXt |dkrutjSt |tt |S|S)Nrcst|Sr/)rr )ZxvrLr"r%sz!Variance.expand..csdt|diS)Nr3)r r r.rLr"r%r|r=r})r?r1rrrFrArrrjmap itertools combinationsrr'rhr) rIrJrOr;reZ variancesZ map_to_covarZ covariancesrkr"rLr%r |s6          zVariance.expandcKs$t|d|}t||d}||S)Nr}r)rIrOr3r4e1e2r"r"r%_eval_rewrite_as_Expectationsz%Variance._eval_rewrite_as_ExpectationcK|ttSr/rTrrrNr"r"r%rzz%Variance._eval_rewrite_as_ProbabilitycKst|jd|jddS)NrFr8)rr?r1rNr"r"r%rPrQz"Variance._eval_rewrite_as_IntegralcCrRr/rSrUr"r"r%rVrWzVariance.evaluate_integralr/) rXrYrZr[r0r rrzrPr\rVr"r"r"r%r=s 0 !   rc@sdeZdZdZdddZddZeddZed d Zdd d Z dd dZ dddZ e Z ddZ dS)r a Symbolic expression for the covariance. Examples ======== >>> from sympy.stats import Covariance >>> from sympy.stats import Normal >>> X = Normal("X", 3, 2) >>> Y = Normal("Y", 0, 1) >>> Z = Normal("Z", 0, 1) >>> W = Normal("W", 0, 1) >>> cexpr = Covariance(X, Y) >>> cexpr Covariance(X, Y) Evaluate the covariance, `X` and `Y` are independent, therefore zero is the result: >>> cexpr.evaluate_integral() 0 Rewrite the covariance expression in terms of expectations: >>> from sympy.stats import Expectation >>> cexpr.rewrite(Expectation) Expectation(X*Y) - Expectation(X)*Expectation(Y) In order to expand the argument, use ``expand()``: >>> from sympy.abc import a, b, c, d >>> Covariance(a*X + b*Y, c*Z + d*W) Covariance(a*X + b*Y, c*Z + d*W) >>> Covariance(a*X + b*Y, c*Z + d*W).expand() a*c*Covariance(X, Z) + a*d*Covariance(W, X) + b*c*Covariance(Y, Z) + b*d*Covariance(W, Y) This class is aware of some properties of the covariance: >>> Covariance(X, X).expand() Variance(X) >>> Covariance(a*X, b*Y).expand() a*b*Covariance(X, Y) NcKst|}t|}|js|jrddlm}||||S|dtjr+t||gtd\}}|dur7t |||}n t|}t ||||}||_ |S)Nr)CrossCovarianceMatrixr9key) rr^r_rrpr r9sortedr rr0r1)r2arg1arg2r3r4rr5r"r"r%r0s   zCovariance.__new__c s|jd}|jd}|j||krt|St|stjSt|s&tjSt||gtd\}}t |t r@t |t r@t ||S| |}| |fdd|D}t |S)Nrr!rc s@g|]\}}D]\}}||tt||gtddiqqS)rr3)r rr )r#reZr1bZr2Zcoeff_rv_list2r3r"r%r&s (z%Covariance.expand..)r?r1rr rrrFrr rArr _expand_single_argumentrrh)rIrJrrZcoeff_rv_list1Zaddendsr"rr%r s$     zCovariance.expandcCst|tr tj|fgSt|tr4g}|jD]}t|tr%|||qt |r1|tj|fq|St|tr?||gSt |rItj|fgSdSr/) rArrrBrr?rrj_get_mul_nonrv_rv_tupler)r2r`Zoutvalrer"r"r%rs        z"Covariance._expand_single_argumentcCsFg}g}|jD]}t|r||q||qt|t|fSr/)r?rrjrrh)r2mr;rkrer"r"r%rs   z"Covariance._get_mul_nonrv_rv_tuplecKs*t|||}t||t||}||Sr/r)rIrrr3r4rrr"r"r%r%sz'Covariance._eval_rewrite_as_ExpectationcKrr/rrIrrr3r4r"r"r%rz*rz'Covariance._eval_rewrite_as_ProbabilitycKst|jd|jd|jddS)Nrr!Fr8)rr?r1rr"r"r%rP-sz$Covariance._eval_rewrite_as_IntegralcCrRr/rSrUr"r"r%rV2rWzCovariance.evaluate_integralr/)rXrYrZr[r0r classmethodrrrrzrPr\rVr"r"r"r%r s ,     r csHeZdZdZdfdd ZddZddd Zdd d Zdd d ZZ S)Momenta Symbolic class for Moment Examples ======== >>> from sympy import Symbol, Integral >>> from sympy.stats import Normal, Expectation, Probability, Moment >>> mu = Symbol('mu', real=True) >>> sigma = Symbol('sigma', real=True, positive=True) >>> X = Normal('X', mu, sigma) >>> M = Moment(X, 3, 1) To evaluate the result of Moment use `doit`: >>> M.doit() mu**3 - 3*mu**2 + 3*mu*sigma**2 + 3*mu - 3*sigma**2 - 1 Rewrite the Moment expression in terms of Expectation: >>> M.rewrite(Expectation) Expectation((X - 1)**3) Rewrite the Moment expression in terms of Probability: >>> M.rewrite(Probability) Integral((x - 1)**3*Probability(Eq(X, x)), (x, -oo, oo)) Rewrite the Moment expression in terms of Integral: >>> M.rewrite(Integral) Integral(sqrt(2)*(X - 1)**3*exp(-(X - mu)**2/(2*sigma**2))/(2*sqrt(pi)*sigma), (X, -oo, oo)) rNc sNt|}t|}t|}|durt|}t|||||St||||Sr/rsuperr0)r2Xncr3r4 __class__r"r%r0YszMoment.__new__cK|tjdi|SNr"rTrr>rIrJr"r"r%r>crQz Moment.doitcKst||||Sr/rrIrrrr3r4r"r"r%rfsz#Moment._eval_rewrite_as_ExpectationcKrr/rrr"r"r%rzirz#Moment._eval_rewrite_as_ProbabilitycKrr/rTrrrr"r"r%rPlrz Moment._eval_rewrite_as_Integral)rN rXrYrZr[r0r>rrzrP __classcell__r"r"rr%r6s"  rcsHeZdZdZd fdd ZddZd ddZd d d Zd d d ZZ S) CentralMomenta1 Symbolic class Central Moment Examples ======== >>> from sympy import Symbol, Integral >>> from sympy.stats import Normal, Expectation, Probability, CentralMoment >>> mu = Symbol('mu', real=True) >>> sigma = Symbol('sigma', real=True, positive=True) >>> X = Normal('X', mu, sigma) >>> CM = CentralMoment(X, 4) To evaluate the result of CentralMoment use `doit`: >>> CM.doit().simplify() 3*sigma**4 Rewrite the CentralMoment expression in terms of Expectation: >>> CM.rewrite(Expectation) Expectation((X - Expectation(X))**4) Rewrite the CentralMoment expression in terms of Probability: >>> CM.rewrite(Probability) Integral((x - Integral(x*Probability(True), (x, -oo, oo)))**4*Probability(Eq(X, x)), (x, -oo, oo)) Rewrite the CentralMoment expression in terms of Integral: >>> CM.rewrite(Integral) Integral(sqrt(2)*(X - Integral(sqrt(2)*X*exp(-(X - mu)**2/(2*sigma**2))/(2*sqrt(pi)*sigma), (X, -oo, oo)))**4*exp(-(X - mu)**2/(2*sigma**2))/(2*sqrt(pi)*sigma), (X, -oo, oo)) Nc sBt|}t|}|durt|}t||||St|||Sr/r)r2rrr3r4rr"r%r0s zCentralMoment.__new__cKrrrrr"r"r%r>rQzCentralMoment.doitcKs.t||fi|}t||||fi|tSr/)rrrT)rIrrr3r4Zmur"r"r%rsz*CentralMoment._eval_rewrite_as_ExpectationcKrr/rrIrrr3r4r"r"r%rzrz*CentralMoment._eval_rewrite_as_ProbabilitycKrr/rrr"r"r%rPrz'CentralMoment._eval_rewrite_as_Integralr/rr"r"rr%rps"  r).rZsympyrrrrrrrr r rir Zsympy.core.compatibilityr Zsympy.core.parametersr Zsympy.core.sympifyrZsympy.core.relationalrZsympy.logic.boolalgrZ sympy.statsrrZsympy.stats.rvrrrrrrrrrr__all__registerr-rrrr rrr"r"r"r%s,0     0  `@q :