o *ήcTK@sdZddlmZddlmZddlmZddlmZddl m Z ddl m Z ddl mZdd lmZdd lmZmZdd lmZmZdd lmZdd lmZddlmZddlmZddlm Z ddl!m"Z"ddl#m$Z$m%Z%ddl&m'Z'm(Z(ddl)m*Z*m+Z+gdZ,ddZ-Gddde'Z.e j/fddZ0Gddde'Z1ddZ2Gd d!d!e'Z3d"d#Z4Gd$d%d%e'Z5d&d'Z6Gd(d)d)e'Z7d*d+Z8Gd,d-d-e'Z9d.d/Z:Gd0d1d1e'Z;d2d3ZGd8d9d9e'Z?d:d;Z@Gdd?ZBd@S)Azn Contains ======== FlorySchulz Geometric Hermite Logarithmic NegativeBinomial Poisson Skellam YuleSimon Zeta )Sum)Basic)Lambda)I)Eq)S)Dummy)sympify)binomial factorial)explog)floor)sqrt) Piecewise)besseli)beta)hyper)polylogzeta)SingleDiscreteDistributionSingleDiscretePSpace) _value_check is_random) FlorySchulz GeometricHermite LogarithmicNegativeBinomialPoissonSkellam YuleSimonZetacOslttt|}||}|ddr|j|t||}tdd|Dr3ddlm}m }||||}|j S)NcheckTcss|]}t|VqdSN)r).0argr'9szrv..r)CompoundPSpaceCompoundDistribution) listmapr popr#ranysympy.stats.compound_rvr*r+value)symbolclsargskwargsdistpspacer*r+r'r'r(rv3s   r8c@s6eZdZdZejfddZeddZe ddZ dS) DiscreteDistributionHandmade)pdfcCst|||Sr$)r__new__)r3r:setr'r'r(r;Bsz$DiscreteDistributionHandmade.__new__cCs |jdSN)r4)selfr'r'r(r<Es z DiscreteDistributionHandmade.setcCs>td}t||||j|jf}tt|dtjkddS)Nxr>z&The pdf is incorrect on the given set.) rr_inf_supdoitrrrfalse)r:r<r@valr'r'r(r#Isz"DiscreteDistributionHandmade.checkN) __name__ __module__ __qualname__ _argnamesrIntegersr;propertyr< staticmethodr#r'r'r'r(r9?s r9cKsNt|}t|||fd}t||}|dd|d<t|jt||fi|S)af Create a Discrete Random Variable given the following: Parameters ========== symbol : Symbol Represents name of the random variable. density : Expression containing symbol Represents probability density function. set : set Represents the region where the pdf is valid, by default is real line. check : bool If True, it will check whether the given density integrates to 1 over the given set. If False, it will not perform this check. Default is False. Examples ======== >>> from sympy.stats import DiscreteRV, P, E >>> from sympy import Rational, Symbol >>> x = Symbol('x') >>> n = 10 >>> density = Rational(1, 10) >>> X = DiscreteRV(x, density, set=set(range(n))) >>> E(X) 9/2 >>> P(X>3) 3/5 Returns ======= RandomSymbol rTr#F)r r as_relationalrr.r8namer9)r2densityr<r5r:r'r'r( DiscreteRVQs & rQc@:eZdZdZejZeddZddZ ddZ dd Z d S) FlorySchulzDistributionacCstd|k|dkfddS)Nrr>za must be between 0 and 1rrTr'r'r(r#zFlorySchulzDistribution.checkcCs"|j}|d|d||dSNr>rT)r?krUr'r'r(r:zFlorySchulzDistribution.pdfcCs6|j}|dtt|d|dtt|dSrX)rUr rr?trUr'r'r(_characteristic_functions0z0FlorySchulzDistribution._characteristic_functioncCs.|j}|dt|d|dt|dSrX)rUr r\r'r'r(_moment_generating_function(z3FlorySchulzDistribution._moment_generating_functionN rFrGrHrIrNaturalsr<rLr#r:r^r_r'r'r'r(rSs  rScC t|t|S)a Create a discrete random variable with a FlorySchulz distribution. The density of the FlorySchulz distribution is given by .. math:: f(k) := (a^2) k (1 - a)^{k-1} Parameters ========== a : A real number between 0 and 1 Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import density, E, variance, FlorySchulz >>> from sympy import Symbol, S >>> a = S.One / 5 >>> z = Symbol("z") >>> X = FlorySchulz("x", a) >>> density(X)(z) (4/5)**(z - 1)*z/25 >>> E(X) 9 >>> variance(X) 40 References ========== https://en.wikipedia.org/wiki/Flory%E2%80%93Schulz_distribution )r8rS)rOrUr'r'r(rs ,rc@rR) GeometricDistributionpcCstd|k|dkfddS)Nrr>zp must be between 0 and 1rVrer'r'r(r#rWzGeometricDistribution.checkcCsd|j|d|jSr=rer?rZr'r'r(r:szGeometricDistribution.pdfcCs.|j}|tt|dd|tt|Sr=)rfr rr?r]rfr'r'r(r^r`z.GeometricDistribution._characteristic_functioncCs&|j}|t|dd|t|Sr=)rfr rhr'r'r(r_ z1GeometricDistribution._moment_generating_functionNrar'r'r'r(rds  rdcCrc)a Create a discrete random variable with a Geometric distribution. Explanation =========== The density of the Geometric distribution is given by .. math:: f(k) := p (1 - p)^{k - 1} Parameters ========== p : A probability between 0 and 1 Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import Geometric, density, E, variance >>> from sympy import Symbol, S >>> p = S.One / 5 >>> z = Symbol("z") >>> X = Geometric("x", p) >>> density(X)(z) (4/5)**(z - 1)/5 >>> E(X) 5 >>> variance(X) 20 References ========== .. [1] https://en.wikipedia.org/wiki/Geometric_distribution .. [2] http://mathworld.wolfram.com/GeometricDistribution.html )r8rdrOrfr'r'r(r 1rc@rR) HermiteDistributiona1a2cCst|jdt|jddS)NzParameter a1 must be >= 0.zParameter a2 must be >= 0.)ris_nonnegativermr'r'r(r#s zHermiteDistribution.checkcCsx|j|j}}t|| }tddd}||d|||}t|d|t|}|t|||d|dfS)NjT)integerrYr)rnror rr rrC)r?rZrnroterm1rqnumdenr'r'r(r:s   zHermiteDistribution.pdfcCs>|j|j}}|t|d}|td|d}t||SNr>rY)rnror r?r]rnrorsterm2r'r'r(r_&s z/HermiteDistribution._moment_generating_functioncCsF|j|j}}|tt|d}|tdt|d}t||Srv)rnror rrwr'r'r(r^,s z,HermiteDistribution._characteristic_functionN) rFrGrHrIr Naturals0r<rLr#r:r_r^r'r'r'r(rls  rlcCt|t||S)a Create a discrete random variable with a Hermite distribution. Explanation =========== The density of the Hermite distribution is given by .. math:: f(x):= e^{-a_1 -a_2}\sum_{j=0}^{\left \lfloor x/2 \right \rfloor} \frac{a_{1}^{x-2j}a_{2}^{j}}{(x-2j)!j!} Parameters ========== a1 : A Positive number greater than equal to 0. a2 : A Positive number greater than equal to 0. Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import Hermite, density, E, variance >>> from sympy import Symbol >>> a1 = Symbol("a1", positive=True) >>> a2 = Symbol("a2", positive=True) >>> x = Symbol("x") >>> H = Hermite("H", a1=5, a2=4) >>> density(H)(2) 33*exp(-9)/2 >>> E(H) 13 >>> variance(H) 21 References ========== .. [1] https://en.wikipedia.org/wiki/Hermite_distribution )r8rl)rOrnror'r'r(r2s4rc@rR) LogarithmicDistributionrecCst|dk|dkfddS)Nrr>p should be between 0 and 1rVrer'r'r(r#qrWzLogarithmicDistribution.checkcCs"|j}d|||td|S)Nr>)rfr )r?rZrfr'r'r(r:ur[zLogarithmicDistribution.pdfcCs*|j}td|tt|td|Sr=)rfr r rrhr'r'r(r^ys$z0LogarithmicDistribution._characteristic_functioncCs&|j}td|t|td|Sr=)rfr r rhr'r'r(r_}riz3LogarithmicDistribution._moment_generating_functionNrar'r'r'r(r{ls  r{cCrc)a Create a discrete random variable with a Logarithmic distribution. Explanation =========== The density of the Logarithmic distribution is given by .. math:: f(k) := \frac{-p^k}{k \ln{(1 - p)}} Parameters ========== p : A value between 0 and 1 Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import Logarithmic, density, E, variance >>> from sympy import Symbol, S >>> p = S.One / 5 >>> z = Symbol("z") >>> X = Logarithmic("x", p) >>> density(X)(z) -1/(5**z*z*log(4/5)) >>> E(X) -1/(-4*log(5) + 8*log(2)) >>> variance(X) -1/((-4*log(5) + 8*log(2))*(-2*log(5) + 4*log(2))) + 1/(-64*log(2)*log(5) + 64*log(2)**2 + 16*log(5)**2) - 10/(-32*log(5) + 64*log(2)) References ========== .. [1] https://en.wikipedia.org/wiki/Logarithmic_distribution .. [2] http://mathworld.wolfram.com/LogarithmicDistribution.html )r8r{rjr'r'r(rrkrc@rR) NegativeBinomialDistributionrrfcCs(t|dkdt|dk|dkfddS)Nrzr should be positiver>r|rVrr'r'r(r#sz"NegativeBinomialDistribution.checkcCs2|j}|j}t||d|d||||Sr=)rrfr )r?rZrrfr'r'r(r:s&z NegativeBinomialDistribution.pdfcCs,|j}|j}d|d|tt||Sr=)rrfr rr?r]rrfr'r'r(r^s z5NegativeBinomialDistribution._characteristic_functioncCs(|j}|j}d|d|t||Sr=)rrfr rr'r'r(r_sz8NegativeBinomialDistribution._moment_generating_functionN rFrGrHrIrryr<rLr#r:r^r_r'r'r'r(r~s  r~cCrz)a Create a discrete random variable with a Negative Binomial distribution. Explanation =========== The density of the Negative Binomial distribution is given by .. math:: f(k) := \binom{k + r - 1}{k} (1 - p)^r p^k Parameters ========== r : A positive value p : A value between 0 and 1 Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import NegativeBinomial, density, E, variance >>> from sympy import Symbol, S >>> r = 5 >>> p = S.One / 5 >>> z = Symbol("z") >>> X = NegativeBinomial("x", r, p) >>> density(X)(z) 1024*binomial(z + 4, z)/(3125*5**z) >>> E(X) 5/4 >>> variance(X) 25/16 References ========== .. [1] https://en.wikipedia.org/wiki/Negative_binomial_distribution .. [2] http://mathworld.wolfram.com/NegativeBinomialDistribution.html )r8r~)rOrrfr'r'r(rs3rc@rR) PoissonDistributionlamdacCt|dkddS)NrzLambda must be positiverVrr'r'r(r#zPoissonDistribution.checkcCs|j|t|t|j Sr$)rr r rgr'r'r(r:zPoissonDistribution.pdfcCst|jtt|dSr=)r rrr?r]r'r'r(r^z,PoissonDistribution._characteristic_functioncCst|jt|dSr=)r rrr'r'r(r_sz/PoissonDistribution._moment_generating_functionNrr'r'r'r(r s  rcCrc)ad Create a discrete random variable with a Poisson distribution. Explanation =========== The density of the Poisson distribution is given by .. math:: f(k) := \frac{\lambda^{k} e^{- \lambda}}{k!} Parameters ========== lamda : Positive number, a rate Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import Poisson, density, E, variance >>> from sympy import Symbol, simplify >>> rate = Symbol("lambda", positive=True) >>> z = Symbol("z") >>> X = Poisson("x", rate) >>> density(X)(z) lambda**z*exp(-lambda)/factorial(z) >>> E(X) lambda >>> simplify(variance(X)) lambda References ========== .. [1] https://en.wikipedia.org/wiki/Poisson_distribution .. [2] http://mathworld.wolfram.com/PoissonDistribution.html )r8r)rOrr'r'r(r rkrc@BeZdZdZejZeddZddZ ddZ dd Z d d Z d S) SkellamDistributionmu1mu2cCs t|dkdt|dkddS)NrzParameter mu1 must be >= 0zParameter mu2 must be >= 0rVrr'r'r(r#\szSkellamDistribution.checkcCsJ|j|j}}t|| |||d}t|dt||}||S)NrY)rrr rr)r?rZrrrsrxr'r'r(r:aszSkellamDistribution.pdfcCstd)Nz-Skellam doesn't have closed form for the CDF.)NotImplementedErrorr?r@r'r'r(_cdfgszSkellamDistribution._cdfcCs>|j|j}}t|| |tt||tt |Sr$)rrr rr?r]rrr'r'r(r^ks0z,SkellamDistribution._characteristic_functioncCs6|j|j}}t|| |t||t| Sr$)rrr rr'r'r(r_os(z/SkellamDistribution._moment_generating_functionN) rFrGrHrIrrJr<rLr#r:rr^r_r'r'r'r(rXs  rcCrz)a Create a discrete random variable with a Skellam distribution. Explanation =========== The Skellam is the distribution of the difference N1 - N2 of two statistically independent random variables N1 and N2 each Poisson-distributed with respective expected values mu1 and mu2. The density of the Skellam distribution is given by .. math:: f(k) := e^{-(\mu_1+\mu_2)}(\frac{\mu_1}{\mu_2})^{k/2}I_k(2\sqrt{\mu_1\mu_2}) Parameters ========== mu1 : A non-negative value mu2 : A non-negative value Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import Skellam, density, E, variance >>> from sympy import Symbol, pprint >>> z = Symbol("z", integer=True) >>> mu1 = Symbol("mu1", positive=True) >>> mu2 = Symbol("mu2", positive=True) >>> X = Skellam("x", mu1, mu2) >>> pprint(density(X)(z), use_unicode=False) z - 2 /mu1\ -mu1 - mu2 / _____ _____\ |---| *e *besseli\z, 2*\/ mu1 *\/ mu2 / \mu2/ >>> E(X) mu1 - mu2 >>> variance(X).expand() mu1 + mu2 References ========== .. [1] https://en.wikipedia.org/wiki/Skellam_distribution )r8r)rOrrr'r'r(r ts8r c@r) YuleSimonDistributionrhocCr)Nrzrho should be positiverVrr'r'r(r#rzYuleSimonDistribution.checkcCs|j}|t||dSr=)rr)r?rZrr'r'r(r:szYuleSimonDistribution.pdfcCs.tdt|tt||jd|dkfdS)Nr>rM)rrrrrr'r'r(rs.zYuleSimonDistribution._cdfcCs8|j}|td|dftt|tt||dSN)r>r>rYr>)rrr rr?r]rr'r'r(r^s2z.YuleSimonDistribution._characteristic_functioncCs0|j}|td|dft|t||dSr)rrr rr'r'r(r_s*z1YuleSimonDistribution._moment_generating_functionN) rFrGrHrIrrbr<rLr#r:rr^r_r'r'r'r(rs  rcCrc)a Create a discrete random variable with a Yule-Simon distribution. Explanation =========== The density of the Yule-Simon distribution is given by .. math:: f(k) := \rho B(k, \rho + 1) Parameters ========== rho : A positive value Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import YuleSimon, density, E, variance >>> from sympy import Symbol, simplify >>> p = 5 >>> z = Symbol("z") >>> X = YuleSimon("x", p) >>> density(X)(z) 5*beta(z, 6) >>> simplify(E(X)) 5/4 >>> simplify(variance(X)) 25/48 References ========== .. [1] https://en.wikipedia.org/wiki/Yule%E2%80%93Simon_distribution )r8r)rOrr'r'r(r! 0r!c@rR) ZetaDistributionscCr)Nr>zs should be greater than 1rVrr'r'r(r#rzZetaDistribution.checkcCs|j}d||t|Sr=)rr)r?rZrr'r'r(r:szZetaDistribution.pdfcCst|jtt|t|jSr$)rrr rrrr'r'r(r^ rz)ZetaDistribution._characteristic_functioncCst|jt|t|jSr$)rrr rrr'r'r(r_rz,ZetaDistribution._moment_generating_functionNrar'r'r'r(rs  rcCrc)a Create a discrete random variable with a Zeta distribution. Explanation =========== The density of the Zeta distribution is given by .. math:: f(k) := \frac{1}{k^s \zeta{(s)}} Parameters ========== s : A value greater than 1 Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import Zeta, density, E, variance >>> from sympy import Symbol >>> s = 5 >>> z = Symbol("z") >>> X = Zeta("x", s) >>> density(X)(z) 1/(z**5*zeta(5)) >>> E(X) pi**4/(90*zeta(5)) >>> variance(X) -pi**8/(8100*zeta(5)**2) + zeta(3)/zeta(5) References ========== .. [1] https://en.wikipedia.org/wiki/Zeta_distribution )r8r)rOrr'r'r(r"rr"N)C__doc__sympy.concrete.summationsrsympy.core.basicrsympy.core.functionrsympy.core.numbersrsympy.core.relationalrsympy.core.singletonrsympy.core.symbolrsympy.core.sympifyr (sympy.functions.combinatorial.factorialsr r &sympy.functions.elementary.exponentialr r #sympy.functions.elementary.integersr(sympy.functions.elementary.miscellaneousr$sympy.functions.elementary.piecewisersympy.functions.special.besselr&sympy.functions.special.beta_functionsrsympy.functions.special.hyperr&sympy.functions.special.zeta_functionsrrsympy.stats.drvrrsympy.stats.rvrr__all__r8r9rJrQrSrrdrrlrr{rr~rrrrr rr!rr"r'r'r'r(sT                128:798>6