a амєz_..)Zfree_symbols┌len┌next┌iter┌any)┌x┌atomsr%r%r&┌_sr/cCsdS)NTr%йr-r%r%r&r/!sc@s8eZdZdZdddДZddДZdddДZeZd d ДZdS) 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|Г}|durtа||б}nt|Г}tа|||б}||_|Sr")rr┌__new__┌ _condition)┌clsZprob┌ condition┌kwargs┌objr%r%r&r1>szProbability.__new__csф|jd}|jЙ|аddб}|аddб}t|tГr\tj|j|jdИ|dНjfi|дОS|а t бrzt|Гj|И|dНStИt Гr·t|Г}t|Гdkr╬|dИkr╬ddlm}||а|бjfi|дОddГStЗfdd Д|DГГrюt|ИГSt|ГабSИduРr tИttfГРs td ИГВИdkРs6|tjuРr.z4%s is not a relational or combination of relationals)r7┌doit)┌argsr2┌get┌ isinstancerr┌One┌funcr>┌hasrrZprobabilityrrr)Zsympy.stats.frv_typesr;r,rrr┌ ValueError┌false┌Zero┌truerrr┌hasattr)┌self┌hintsr4r7r8Zcondrvr;┌resultr%r=r&r>Hs\ zProbability.doitcKs|j||dНjddНS)Nйr4T)r8йrCr>йrJ┌argr4r5r%r%r&┌_eval_rewrite_as_Integral}sz%Probability._eval_rewrite_as_IntegralcCs|аtбабSr"й┌rewriter r>йrJr%r%r&┌evaluate_integralВszProbability.evaluate_integral)N)N) ┌__name__┌ __module__┌__qualname__┌__doc__r1r>rQ┌_eval_rewrite_as_SumrUr%r%r%r&r&s 5 rc@sNeZdZdZdddДZddДZddДZdd d ДZdddДZeZ d dДZ e ZdS)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|Г}|jr$ddlm}|||ГS|durFt|Гs8|Stа||б}nt|Г}tа|||б}||_|S)Nr)┌ExpectationMatrix)r┌ is_Matrix┌-sympy.stats.symbolic_multivariate_probabilityr[rrr1r2)r3┌exprr4r5r[r6r%r%r&r1╠s zExpectation.__new__cs╚|jd}|jЙt|Гs|St|tГr@tаЗfddД|jDГбSt|Г}t|tГrltаЗfddД|jDГбSt|tГr─g}g}|jD]"}t|ГrЬ|а|бqД|а|бqДtа|бt tа|бИdНS|S)Nrc3s|]}t|ИdНабVqdSйrMNйrrйr#┌arMr%r&r'уs z%Expectation.expand..c3s|]}t|ИdНабVqdSr_r`rarMr%r&r'шs rM) r?r2rrAr┌fromiter┌_expandr┌appendr)rJrKr^Zexpand_exprr<┌nonrvrbr%rMr&r█s, zExpectation.expandcsJИаddб}ИjЙИjd}Иаddб}Иаddб}|rJ|jfiИдО}t|Гr\t|tГr`|S|rАИаddб}t|И||dНS|аt бrЪt |Га|ИбSИdur╛Иаt |ИГбjfiИдОS|jrрtЗЗЗfd d Д|jDГОS|jrЇ|аtбrЇ|St |ГtГkРrИа|бSt |Гj||dН}t|dГРrB|РrB|jfiИдОS|SdS) N┌deepTrr7Fr8┌evalf)r7rhcs:g|]2}t|tГs*Иа|ИбjfiИдОn Иа|ИбСqSr%)rArrCr>)r#rPйr4rKrJr%r&┌ s z$Expectation.doit..r9r>)r@r2r?r>rrArrrDrrZcompute_expectationrCrZis_AddrZis_Mulr.rrI)rJrKrgr^r7r8rhrLr%rir&r>ўs: ■ zExpectation.doitcKs|аtб}t|ГdkrtГВt|Гdkr,|S|аб}|jdurFtdГВ|j}|jdа бrjt |jабГ}nt |jdГ}|jjr╕t |а||бtt||Г|Г||jjjj|jjjjfГS|jjr╞tВn6t|а||бtt||Г|Г||jjjj|jjjfГSdS)Nr!rzProbability space not knownZ_1)r.rr)┌NotImplementedError┌poprrE┌symbol┌name┌isupperr ┌lowerZ is_Continuousr ┌replacerr┌domain┌set┌inf┌supZ is_Finiter)rJrPr4r5Zrvsr<rmr%r%r&┌_eval_rewrite_as_Probability"s" 8z(Expectation._eval_rewrite_as_ProbabilitycKs|j||dНjdddНS)NrMFT)rgr8rNrOr%r%r&rQ;sz%Expectation._eval_rewrite_as_IntegralcCs|аtбабSr"rRrTr%r%r&rU@szExpectation.evaluate_integral)N)N)N)rVrWrXrYr1rr>rvrQrZrUZevaluate_sumr%r%r%r&rЖsE + rc@sLeZdZdZdddДZddДZdddДZdd d ДZdddДZeZ d dДZ 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|Г}|jr$ddlm}|||ГS|dur:tа||б}nt|Г}tа|||б}||_|S)Nr)┌VarianceMatrix)rr\r]rwrr1r2)r3rPr4r5rwr6r%r%r&r1vs zVariance.__new__c s |jd}|jЙt|ГstjSt|tГr,|St|tГrШg}|jD]}t|Гr@|а|бq@tt ЗfddД|ГО}ЗfddД}tt |t а|dбГО}||St|tГРrg}g}|jD]&}t|Гr╩|а|бq▓|а|dбq▓t |ГdkrьtjStа|бttа|бИГS|S)Nrcst|ИГабSr")rr)ZxvrMr%r&┌Тr(z!Variance.expand..csdt|dИiОабS)Nщr4)r rr0rMr%r&rxУr(ry)r?r2rrrGrArrre┌map┌ itertools┌combinationsrr)rcr) rJrKrPr<rbZ variancesZmap_to_covarZcovariancesrfr%rMr&rДs4 zVariance.expandcKs$t|d|Г}t||Гd}||S)Nryйr)rJrPr4r5┌e1┌e2r%r%r&┌_eval_rewrite_as_Expectationеsz%Variance._eval_rewrite_as_ExpectationcKs|аtбаtбSr"йrSrrrOr%r%r&rvкsz%Variance._eval_rewrite_as_ProbabilitycKst|jd|jddНS)NrFr9)rr?r2rOr%r%r&rQнsz"Variance._eval_rewrite_as_IntegralcCs|аtбабSr"rRrTr%r%r&rU▓szVariance.evaluate_integral)N)N)N)N)rVrWrXrYr1rrАrvrQrZrUr%r%r%r&rEs0 ! rc@sdeZdZdZdddДZddДZeddДГZed d ДГZdddДZ dd dДZ dddДZeZddДZ dS)r aШ 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) NcKsРt|Г}t|Г}|js|jr4ddlm}||||ГS|аdtjбrVt||gtdН\}}|durnt а |||б}nt|Г}t а ||||б}||_|S)Nr)┌CrossCovarianceMatrixr:й┌key)rr\r]rВrlrr:┌sortedr rr1r2)r3┌arg1┌arg2r4r5rВr6r%r%r&r1уszCovariance.__new__cs║|jd}|jd}|jЙ||kr0t|ИГабSt|Гs>tjSt|ГsLtjSt||gtdН\}}t |t ГrАt |t ГrАt||ИГS|а|абб}|а|аббЙЗЗfddД|DГ}t а|бS)Nrr!rГc s@g|]8\}}ИD]*\}}||tt||gtdНdИiОСqqS)rГr4)r rЕr )r#rbZr1┌bZr2йZcoeff_rv_list2r4r%r&rjs z%Covariance.expand..)r?r2rrrrrGrЕr rArr ┌_expand_single_argumentrrc)rJrKrЖrЗZcoeff_rv_list1Zaddendsr%rЙr&rЎs$ zCovariance.expandcCsЦt|tГrtj|fgSt|tГrhg}|jD]8}t|tГrJ|а|а|ббq*t |Гr*|аtj|fбq*|St|tГr~|а|бgSt |ГrТtj|fgSdSr") rArrrBrr?rre┌_get_mul_nonrv_rv_tupler)r3r^Zoutvalrbr%r%r&rКs z"Covariance._expand_single_argumentcCsFg}g}|jD]"}t|Гr&|а|бq|а|бqtа|бtа|бfSr")r?rrerrc)r3┌mr<rfrbr%r%r&rЛ"s z"Covariance._get_mul_nonrv_rv_tuplecKs*t|||Г}t||Гt||Г}||Sr"r})rJrЖrЗr4r5r~rr%r%r&rА-sz'Covariance._eval_rewrite_as_ExpectationcKs|аtбаtбSr"rБйrJrЖrЗr4r5r%r%r&rv2sz'Covariance._eval_rewrite_as_ProbabilitycKst|jd|jd|jddНS)Nrr!Fr9)rr?r2rНr%r%r&rQ5sz$Covariance._eval_rewrite_as_IntegralcCs|аtбабSr"rRrTr%r%r&rU:szCovariance.evaluate_integral)N)N)N)N)rVrWrXrYr1r┌classmethodrКrЛrАrvrQrZrUr%r%r%r&r ╢s, r csHeZdZdZdЗfddД ZddДZddd ДZdd dДZddd ДZЗZ S)┌MomentaП 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', 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)) rNcsRt|Г}t|Г}t|Г}|durйrJrKr%r%r&r>kszMoment.doitcKst||||ГSr"r}йrJrТrУrФr4r5r%r%r&rАnsz#Moment._eval_rewrite_as_ExpectationcKs|аtбаtбSr"rБrЩr%r%r&rvqsz#Moment._eval_rewrite_as_ProbabilitycKs|аtбаtбSr"йrSrr rЩr%r%r&rQtsz Moment._eval_rewrite_as_Integral)rN)rN)rN)rNй rVrWrXrYr1r>rАrvrQ┌ __classcell__r%r%rХr&rП>s" rПcsHeZdZdZd ЗfddД ZddДZdddДZdd d ДZdddДZЗZ S)┌ CentralMomenta& 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', 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)) NcsFt|Г}t|Г}|dur2t|Г}tГа||||бStГа|||бSdSr"rР)r3rТrУr4r5rХr%r&r1ЫszCentralMoment.__new__cKs|аtбjfi|дОSr"rЧrШr%r%r&r>дszCentralMoment.doitcKs.t||fi|дО}t||||fi|дОаtбSr")rrПrS)rJrТrУr4r5┌mur%r%r&rАзsz*CentralMoment._eval_rewrite_as_ExpectationcKs|аtбаtбSr"rБйrJrТrУr4r5r%r%r&rvлsz*CentralMoment._eval_rewrite_as_ProbabilitycKs|аtбаtбSr"rЪrЯr%r%r&rQоsz'CentralMoment._eval_rewrite_as_Integral)N)N)N)NrЫr%r%rХr&rЭxs" rЭ)5r{Zsympy.concrete.summationsrZsympy.core.addrZsympy.core.exprrZsympy.core.functionrrdZsympy.core.mulrZsympy.core.relationalrZsympy.core.singletonrZsympy.core.symbolr Zsympy.integrals.integralsr Zsympy.logic.boolalgrZsympy.core.parametersrZsympy.core.sortingr Zsympy.core.sympifyrrrZsympy.statsrrZsympy.stats.rvrrrrrrrrrr┌__all__┌registerr/rrrr rПrЭr%r%r%r&┌s<0 `@q :