o *ήc%@sdZddlmZddlmZmZmZmZddlm Z m Z m Z ddl m ZddlmZddlmZddlmZdd lmZmZmZdd lmZmZdd lmZdd lmZdd l m!Z!ddl"m#Z#m$Z$ddl%m&Z&m'Z'ddlm(Z(ddl)m*Z*ddl+m,Z,ddl-m.Z.m/Z/m0Z0ddl1m2Z2ddlm3Z3ddl4m5Z5m6Z6m7Z7m8Z8m9Z9ddl:m;Z;mm?Z?ddl@mAZAddlBmCZCddlDmEZEddlFmGZGddlHmIZImJZJddlKmLZLmMZMejNZOgdZPeMQeGd d!ZRd"d#ZSGd$d%d%eJZTeEeO eOfd&d'ZUGd(d)d)eJZVdd+d,ZWGd-d.d.eJZXd/d0ZYGd1d2d2eJZZd3d4Z[Gd5d6d6eJZ\d7d8Z]Gd9d:d:eJZ^d;d<Z_Gd=d>d>eJZ`d?d@ZaGdAdBdBeJZbdCdDZcGdEdFdFeJZddGdHZeGdIdJdJeJZfdKdLZgGdMdNdNeJZhdOdPZiGdQdRdReJZjdSdTZkdUdVZlGdWdXdXeJZmdYdZZnGd[d\d\eJZod]d^ZpGd_d`d`eJZqdadbZrGdcddddeJZsdedfZtGdgdhdheJZudidjZvGdkdldleJZwddmdnZxGdodpdpeJZydqdrZzGdsdtdteJZ{dudvZ|GdwdxdxeJZ}ddzd{Z~Gd|d}d}eJZd~dZGdddeJZddZGdddeJZddZGdddeJZddZGdddeJZddZGdddeJZddZGdddeJZddZGdddeJZddZGdddeJZddZGdddeJZddZGdddeJZddZGdddeJZddZGdddeJZddZGdddeJZddZGdddeJZddZeZGdddeJZddZGdddeJZddZGdddeJZddÄZGddńdeJZddDŽZGddɄdeJZdd˄ZGdd̈́deJZddτZGddфdeJZddӄZGddՄdeJZddׄZGddلdeJZddۄZGdd݄deJZdd߄ZGdddeJZddZGdddeJZddZGdddeJZddZGdddeJZddZGdddeJZddZdS)a6 Continuous Random Variables - Prebuilt variables Contains ======== Arcsin Benini Beta BetaNoncentral BetaPrime BoundedPareto Cauchy Chi ChiNoncentral ChiSquared Dagum Erlang ExGaussian Exponential ExponentialPower FDistribution FisherZ Frechet Gamma GammaInverse Gumbel Gompertz Kumaraswamy Laplace Levy LogCauchy Logistic LogLogistic LogitNormal LogNormal Lomax Maxwell Moyal Nakagami Normal Pareto PowerFunction QuadraticU RaisedCosine Rayleigh Reciprocal ShiftedGompertz StudentT Trapezoidal Triangular Uniform UniformSum VonMises Wald Weibull WignerSemicircle )exp)atancossintan)besselibesseljbesselk)beta)Sum)Basic)Lambda)IRationalpi)EqNe)S)Dummy)sympify)binomial factorial)Abssignlog)sinh)floor)sqrtMaxMin) Piecewise)asin)erferfcerfierfinvexpint)gamma lowergamma uppergamma)hyper) integrate)And)Interval) MatrixBase)SingleContinuousPSpaceSingleContinuousDistribution) _value_check is_random)5 ContinuousRVArcsinBeniniBetaBetaNoncentral BetaPrime BoundedParetoCauchyChi ChiNoncentral ChiSquaredDagumErlang ExGaussian ExponentialExponentialPower FDistributionFisherZFrechetGamma GammaInverseGompertzGumbel KumaraswamyLaplaceLevy LogCauchyLogistic LogLogistic LogitNormal LogNormalLomaxMaxwellMoyalNakagamiNormalGaussianInversePareto PowerFunction QuadraticU RaisedCosineRayleigh ReciprocalStudentTShiftedGompertz Trapezoidal TriangularUniform UniformSumVonMisesWaldWeibullWignerSemicirclecCstdd|DS)Ncs|]}t|VqdSNr3).0irnz_..)any)xrnrnro_rtcKslttt|}||}|ddr|j|t||}tdd|Dr3ddlm}m }||||}|j S)NcheckTcsrirjrk)rlargrnrnrorprqzrv..r)CompoundPSpaceCompoundDistribution) listmaprpoprvr0rrsympy.stats.compound_rvrxryvalue)symbolclsargskwargsdistpspacerxryrnrnrorvs   rc@s<eZdZdZee efddZeddZe ddZ dS) ContinuousDistributionHandmade)pdfcCst|||Srj)r __new__)rrsetrnrnrorsz&ContinuousDistributionHandmade.__new__cCs |jdSN)rselfrnrnror z"ContinuousDistributionHandmade.setcCs4td}t||||f}tt|dtjkddS)Nrsrz&The pdf on the given set is incorrect.)rr,r2rrfalse)rrrsvalrnrnrorvsz$ContinuousDistributionHandmade.checkN) __name__ __module__ __qualname__ _argnamesr.oorpropertyr staticmethodrvrnrnrnrors rcKsHt|||fd}t||}|dd|d<t|jt||ffi|S)a Create a Continuous Random Variable given the following: Parameters ========== symbol : Symbol Represents name of the random variable. density : Expression containing symbol Represents probability density function. set : set/Interval 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. Returns ======= RandomSymbol Many common continuous random variable types are already implemented. This function should be necessary only very rarely. Examples ======== >>> from sympy import Symbol, sqrt, exp, pi >>> from sympy.stats import ContinuousRV, P, E >>> x = Symbol("x") >>> pdf = sqrt(2)*exp(-x**2/2)/(2*sqrt(pi)) # Normal distribution >>> X = ContinuousRV(x, pdf) >>> E(X) 0 >>> P(X>0) 1/2 rTrvF)r! as_relationalr r|rnamer)rdensityrrrrnrnror4s, r4c@s,eZdZdZeddZddZddZdS) ArcsinDistributionabcCt|j|jSrjr.rrrrnrnrorzArcsinDistribution.setcCs*|j|j}}dtt||||Sr)rrrrrrsrrrnrnrorzArcsinDistribution.pdfcCsN|j|j}}ttj||kfdtt||||t||kftjdfSNT) rrr!rZeror"rrOnerrnrnro_cdfs  &zArcsinDistribution._cdfN)rrrrrrrrrnrnrnrors   rrcCt|t||fS)a Create a Continuous Random Variable with an arcsin distribution. The density of the arcsin distribution is given by .. math:: f(x) := \frac{1}{\pi\sqrt{(x-a)(b-x)}} with :math:`x \in (a,b)`. It must hold that :math:`-\infty < a < b < \infty`. Parameters ========== a : Real number, the left interval boundary b : Real number, the right interval boundary Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import Arcsin, density, cdf >>> from sympy import Symbol >>> a = Symbol("a", real=True) >>> b = Symbol("b", real=True) >>> z = Symbol("z") >>> X = Arcsin("x", a, b) >>> density(X)(z) 1/(pi*sqrt((-a + z)*(b - z))) >>> cdf(X)(z) Piecewise((0, a > z), (2*asin(sqrt((-a + z)/(-a + b)))/pi, b >= z), (1, True)) References ========== .. [1] https://en.wikipedia.org/wiki/Arcsine_distribution )rrrrrrnrnror52r5c@s8eZdZdZeddZeddZddZdd Z d S) BeniniDistributionalphar sigmacC.t|dkdt|dkdt|dkddS)Nr'Shape parameter Alpha must be positive.&Shape parameter Beta must be positive.z'Scale parameter Sigma must be positive.r2rrnrnrorv>zBeniniDistribution.checkcC t|jtSrj)r.rrrrnrnrorD zBeniniDistribution.setcCs`|j|j|j}}}t| t|||t||d||d|t|||SNr)rr rrr)rrsrr rrnrnrorHs(zBeniniDistribution.pdfcCtd)NzIThe moment generating function of the Benini distribution does not exist.NotImplementedErrorrtrnrnro_moment_generating_functionMz.BeniniDistribution._moment_generating_functionN) rrrrrrvrrrrrnrnrnror;s   rcCt|t|||fS)aW Create a Continuous Random Variable with a Benini distribution. The density of the Benini distribution is given by .. math:: f(x) := e^{-\alpha\log{\frac{x}{\sigma}} -\beta\log^2\left[{\frac{x}{\sigma}}\right]} \left(\frac{\alpha}{x}+\frac{2\beta\log{\frac{x}{\sigma}}}{x}\right) This is a heavy-tailed distribution and is also known as the log-Rayleigh distribution. Parameters ========== alpha : Real number, `\alpha > 0`, a shape beta : Real number, `\beta > 0`, a shape sigma : Real number, `\sigma > 0`, a scale Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import Benini, density, cdf >>> from sympy import Symbol, pprint >>> alpha = Symbol("alpha", positive=True) >>> beta = Symbol("beta", positive=True) >>> sigma = Symbol("sigma", positive=True) >>> z = Symbol("z") >>> X = Benini("x", alpha, beta, sigma) >>> D = density(X)(z) >>> pprint(D, use_unicode=False) / / z \\ / z \ 2/ z \ | 2*beta*log|-----|| - alpha*log|-----| - beta*log |-----| |alpha \sigma/| \sigma/ \sigma/ |----- + -----------------|*e \ z z / >>> cdf(X)(z) Piecewise((1 - exp(-alpha*log(z/sigma) - beta*log(z/sigma)**2), sigma <= z), (0, True)) References ========== .. [1] https://en.wikipedia.org/wiki/Benini_distribution .. [2] http://reference.wolfram.com/legacy/v8/ref/BeniniDistribution.html )rr)rrr rrnrnror6QseZdZdZeddZeddZddZdd Z d d Z d S) BetaDistributionrr rrcC t|dkdt|dkddSNrrrrrrnrnrorvzBetaDistribution.checkcCs4|j|j}}||dd||dt||Srrr beta_fnrrsrr rnrnror&zBetaDistribution.pdfcCst|jf|j|jft|Srj)r+rr rrrnrnro_characteristic_functionz)BetaDistribution._characteristic_functioncCst|jf|j|jf|Srj)r+rr rrnrnrorz,BetaDistribution._moment_generating_functionN) rrrrr.rrrvrrrrnrnrnrors   rcCr)a Create a Continuous Random Variable with a Beta distribution. The density of the Beta distribution is given by .. math:: f(x) := \frac{x^{\alpha-1}(1-x)^{\beta-1}} {\mathrm{B}(\alpha,\beta)} with :math:`x \in [0,1]`. Parameters ========== alpha : Real number, `\alpha > 0`, a shape beta : Real number, `\beta > 0`, a shape Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import Beta, density, E, variance >>> from sympy import Symbol, simplify, pprint, factor >>> alpha = Symbol("alpha", positive=True) >>> beta = Symbol("beta", positive=True) >>> z = Symbol("z") >>> X = Beta("x", alpha, beta) >>> D = density(X)(z) >>> pprint(D, use_unicode=False) alpha - 1 beta - 1 z *(1 - z) -------------------------- B(alpha, beta) >>> simplify(E(X)) alpha/(alpha + beta) >>> factor(simplify(variance(X))) alpha*beta/((alpha + beta)**2*(alpha + beta + 1)) References ========== .. [1] https://en.wikipedia.org/wiki/Beta_distribution .. [2] http://mathworld.wolfram.com/BetaDistribution.html )rrrrr rnrnror77r7c@s.eZdZdZeddZeddZddZdS) BetaNoncentralDistributionrr lamdarrcCs.t|dkdt|dkdt|dkddS)Nrrrz/Noncentrality parameter Lambda must be positiverrrnrnrorvrz BetaNoncentralDistribution.checkcCsz|j|j|j}}}td}tt| d|d||||dd||dt|t||||dtfS)Nkrrr) rr rrr rrrr)rrsrr rrrnrnrors* zBetaNoncentralDistribution.pdfN) rrrrr.rrrvrrnrnrnrors    rcCr)at Create a Continuous Random Variable with a Type I Noncentral Beta distribution. The density of the Noncentral Beta distribution is given by .. math:: f(x) := \sum_{k=0}^\infty e^{-\lambda/2}\frac{(\lambda/2)^k}{k!} \frac{x^{\alpha+k-1}(1-x)^{\beta-1}}{\mathrm{B}(\alpha+k,\beta)} with :math:`x \in [0,1]`. Parameters ========== alpha : Real number, `\alpha > 0`, a shape beta : Real number, `\beta > 0`, a shape lamda : Real number, `\lambda \geq 0`, noncentrality parameter Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import BetaNoncentral, density, cdf >>> from sympy import Symbol, pprint >>> alpha = Symbol("alpha", positive=True) >>> beta = Symbol("beta", positive=True) >>> lamda = Symbol("lamda", nonnegative=True) >>> z = Symbol("z") >>> X = BetaNoncentral("x", alpha, beta, lamda) >>> D = density(X)(z) >>> pprint(D, use_unicode=False) oo _____ \ ` \ -lamda \ k ------- \ k + alpha - 1 /lamda\ beta - 1 2 ) z *|-----| *(1 - z) *e / \ 2 / / ------------------------------------------------ / B(k + alpha, beta)*k! /____, k = 0 Compute cdf with specific 'x', 'alpha', 'beta' and 'lamda' values as follows: >>> cdf(BetaNoncentral("x", 1, 1, 1), evaluate=False)(2).doit() 2*exp(1/2) The argument evaluate=False prevents an attempt at evaluation of the sum for general x, before the argument 2 is passed. References ========== .. [1] https://en.wikipedia.org/wiki/Noncentral_beta_distribution .. [2] https://reference.wolfram.com/language/ref/NoncentralBetaDistribution.html )rr)rrr rrnrnror8sDr8c@.eZdZdZeddZedeZddZ dS)BetaPrimeDistributionrcCrrrrrnrnrorvDrzBetaPrimeDistribution.checkrcCs6|j|j}}||dd|| |t||SrrrrnrnrorK(zBetaPrimeDistribution.pdfN rrrrrrvr.rrrrnrnrnrorAs    rcCr)a Create a continuous random variable with a Beta prime distribution. The density of the Beta prime distribution is given by .. math:: f(x) := \frac{x^{\alpha-1} (1+x)^{-\alpha -\beta}}{B(\alpha,\beta)} with :math:`x > 0`. Parameters ========== alpha : Real number, `\alpha > 0`, a shape beta : Real number, `\beta > 0`, a shape Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import BetaPrime, density >>> from sympy import Symbol, pprint >>> alpha = Symbol("alpha", positive=True) >>> beta = Symbol("beta", positive=True) >>> z = Symbol("z") >>> X = BetaPrime("x", alpha, beta) >>> D = density(X)(z) >>> pprint(D, use_unicode=False) alpha - 1 -alpha - beta z *(z + 1) ------------------------------- B(alpha, beta) References ========== .. [1] https://en.wikipedia.org/wiki/Beta_prime_distribution .. [2] http://mathworld.wolfram.com/BetaPrimeDistribution.html )rrrrnrnror9O1r9c@0eZdZdZeddZeddZddZdS) BoundedParetoDistributionrleftrightcCrrjr.rrrrnrnrorrzBoundedParetoDistribution.setcCs*t|jdt|jdt||kddS)NzShape must be positive.zLeft value should be positive.z"Right should be greater than left.)r2 is_positiverrnrnrorvs  zBoundedParetoDistribution.checkcCsH|j|j|j}}}||||| d}d|||}||Srr)rrsrrrnumdenrnrnrorszBoundedParetoDistribution.pdfN rrrrrrrrvrrnrnrnror   rcCr)a- Create a continuous random variable with a Bounded Pareto distribution. The density of the Bounded Pareto distribution is given by .. math:: f(x) := \frac{\alpha L^{\alpha}x^{-\alpha-1}}{1-(\frac{L}{H})^{\alpha}} Parameters ========== alpha : Real Number, `\alpha > 0` Shape parameter left : Real Number, `left > 0` Location parameter right : Real Number, `right > left` Location parameter Examples ======== >>> from sympy.stats import BoundedPareto, density, cdf, E >>> from sympy import symbols >>> L, H = symbols('L, H', positive=True) >>> X = BoundedPareto('X', 2, L, H) >>> x = symbols('x') >>> density(X)(x) 2*L**2/(x**3*(1 - L**2/H**2)) >>> cdf(X)(x) Piecewise((-H**2*L**2/(x**2*(H**2 - L**2)) + H**2/(H**2 - L**2), L <= x), (0, True)) >>> E(X).simplify() 2*H*L/(H + L) Returns ======= RandomSymbol References ========== .. [1] https://en.wikipedia.org/wiki/Pareto_distribution#Bounded_Pareto_distribution )rr)rrrrrnrnror:s-r:c@DeZdZdZeddZddZddZdd Zd d Z d d Z dS)CauchyDistributionx0r(cCt|dkdt|jddS)Nr'Scale parameter Gamma must be positive. Location parameter must be real.r2is_realrrnrnrorvzCauchyDistribution.checkcCs&dt|jd||j|jdSNrr)rr(rrrsrnrnrors&zCauchyDistribution.pdfcCs,|j|j}}dtt|||tjSr)rr(rrrHalf)rrsrr(rnrnrorszCauchyDistribution._cdfcCs t|jt||jt|Srj)rrrr(rrrnrnrors z+CauchyDistribution._characteristic_functioncCr)NzJThe moment generating function for the Cauchy distribution does not exist.rrrnrnrorrz.CauchyDistribution._moment_generating_functioncCs|j|jtt|tjSrj)rr(rrrrrprnrnro _quantilerzCauchyDistribution._quantileN rrrrrrvrrrrrrnrnrnrors  rcCr)aH Create a continuous random variable with a Cauchy distribution. The density of the Cauchy distribution is given by .. math:: f(x) := \frac{1}{\pi \gamma [1 + {(\frac{x-x_0}{\gamma})}^2]} Parameters ========== x0 : Real number, the location gamma : Real number, `\gamma > 0`, a scale Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import Cauchy, density >>> from sympy import Symbol >>> x0 = Symbol("x0") >>> gamma = Symbol("gamma", positive=True) >>> z = Symbol("z") >>> X = Cauchy("x", x0, gamma) >>> density(X)(z) 1/(pi*gamma*(1 + (-x0 + z)**2/gamma**2)) References ========== .. [1] https://en.wikipedia.org/wiki/Cauchy_distribution .. [2] http://mathworld.wolfram.com/CauchyDistribution.html )rr)rrr(rnrnror;s+r;c@s>eZdZdZeddZedeZddZ ddZ d d Z d S) ChiDistributionrcCt|dkdt|jddSNr2Number of degrees of freedom (k) must be positive.4Number of degrees of freedom (k) must be an integer.r2 is_integerrrnrnrorvrzChiDistribution.checkrcCs@dd|jd||jdt|d dt|jdSNrr)rrr(rrnrnrors@zChiDistribution.pdfcCs|j}t|dftjf|d d}t|tdt|ddt|d}t|ddftddf|d d}|||SNrr)rr+rrrrr(rrrrpart_1part_2part_3rnrnror"s  ,( z(ChiDistribution._characteristic_functioncCs|j}t|dftjf|dd}|tdt|ddt|d}t|ddftddf|dd}|||Sr)rr+rrrr(rrnrnror*s (( z+ChiDistribution._moment_generating_functionN) rrrrrrvr.rrrrrrnrnrnrors   rcCt|t|fS)a] Create a continuous random variable with a Chi distribution. The density of the Chi distribution is given by .. math:: f(x) := \frac{2^{1-k/2}x^{k-1}e^{-x^2/2}}{\Gamma(k/2)} with :math:`x \geq 0`. Parameters ========== k : Positive integer, The number of degrees of freedom Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import Chi, density, E >>> from sympy import Symbol, simplify >>> k = Symbol("k", integer=True) >>> z = Symbol("z") >>> X = Chi("x", k) >>> density(X)(z) 2**(1 - k/2)*z**(k - 1)*exp(-z**2/2)/gamma(k/2) >>> simplify(E(X)) sqrt(2)*gamma(k/2 + 1/2)/gamma(k/2) References ========== .. [1] https://en.wikipedia.org/wiki/Chi_distribution .. [2] http://mathworld.wolfram.com/ChiDistribution.html )rrrrrnrnror<2s.r<c@r)ChiNoncentralDistributionrlcCs,t|dkdt|jdt|dkddS)Nrrrz(Shift parameter Lambda must be positive.rrrnrnrorvis zChiNoncentralDistribution.checkrcCsZ|j|j}}t|d|d d||||||dt|dd||Sr)rrrr)rrsrrrnrnrorqsLzChiNoncentralDistribution.pdfNrrnrnrnrorfs    rcCr)a0 Create a continuous random variable with a non-central Chi distribution. Explanation =========== The density of the non-central Chi distribution is given by .. math:: f(x) := \frac{e^{-(x^2+\lambda^2)/2} x^k\lambda} {(\lambda x)^{k/2}} I_{k/2-1}(\lambda x) with `x \geq 0`. Here, `I_\nu (x)` is the :ref:`modified Bessel function of the first kind `. Parameters ========== k : A positive Integer, $k > 0$ The number of degrees of freedom. lambda : Real number, `\lambda > 0` Shift parameter. Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import ChiNoncentral, density >>> from sympy import Symbol >>> k = Symbol("k", integer=True) >>> l = Symbol("l") >>> z = Symbol("z") >>> X = ChiNoncentral("x", k, l) >>> density(X)(z) l*z**k*exp(-l**2/2 - z**2/2)*besseli(k/2 - 1, l*z)/(l*z)**(k/2) References ========== .. [1] https://en.wikipedia.org/wiki/Noncentral_chi_distribution )rrrrrrnrnror=urr=c@sFeZdZdZeddZedeZddZ ddZ d d Z d d Z d S)ChiSquaredDistributionrcCrrrrrnrnrorvrzChiSquaredDistribution.checkrcCs@|j}dd|dt|d||ddt| dSr)rr(rrrsrrnrnror:zChiSquaredDistribution.pdfcCs8|j}ttjt|dt|d|d|dkfdS)Nrrr)rr!rrr(r)r rnrnrors *zChiSquaredDistribution._cdfcCsddt||j dSr)rrrrnrnrorsz/ChiSquaredDistribution._characteristic_functioncCsdd||j dSrrrrnrnrorsz2ChiSquaredDistribution._moment_generating_functionN) rrrrrrvr.rrrrrrrnrnrnror s   r cCr)a" Create a continuous random variable with a Chi-squared distribution. Explanation =========== The density of the Chi-squared distribution is given by .. math:: f(x) := \frac{1}{2^{\frac{k}{2}}\Gamma\left(\frac{k}{2}\right)} x^{\frac{k}{2}-1} e^{-\frac{x}{2}} with :math:`x \geq 0`. Parameters ========== k : Positive integer The number of degrees of freedom. Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import ChiSquared, density, E, variance, moment >>> from sympy import Symbol >>> k = Symbol("k", integer=True, positive=True) >>> z = Symbol("z") >>> X = ChiSquared("x", k) >>> density(X)(z) z**(k/2 - 1)*exp(-z/2)/(2**(k/2)*gamma(k/2)) >>> E(X) k >>> variance(X) 2*k >>> moment(X, 3) k**3 + 6*k**2 + 8*k References ========== .. [1] https://en.wikipedia.org/wiki/Chi_squared_distribution .. [2] http://mathworld.wolfram.com/Chi-SquaredDistribution.html )rr rrnrnror>s8r>c@6eZdZdZedeZeddZddZ ddZ d S) DagumDistributionrrrrcCr)Nrz#Shape parameter p must be positive.z#Shape parameter a must be positive.#Scale parameter b must be positive.rrrnrnrorv rzDagumDistribution.checkcCsJ|j|j|j}}}||||||||||d|dSrrrrsrrrrnrnror4zDagumDistribution.pdfcCsH|j|j|j}}}ttjt||| | |dkftjdfSNrT)rrrr!rrrrrnrnrors&zDagumDistribution._cdfN rrrrr.rrrrvrrrnrnrnror s   r cCr)aK Create a continuous random variable with a Dagum distribution. Explanation =========== The density of the Dagum distribution is given by .. math:: f(x) := \frac{a p}{x} \left( \frac{\left(\tfrac{x}{b}\right)^{a p}} {\left(\left(\tfrac{x}{b}\right)^a + 1 \right)^{p+1}} \right) with :math:`x > 0`. Parameters ========== p : Real number `p > 0`, a shape. a : Real number `a > 0`, a shape. b : Real number `b > 0`, a scale. Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import Dagum, density, cdf >>> from sympy import Symbol >>> p = Symbol("p", positive=True) >>> a = Symbol("a", positive=True) >>> b = Symbol("b", positive=True) >>> z = Symbol("z") >>> X = Dagum("x", p, a, b) >>> density(X)(z) a*p*(z/b)**(a*p)*((z/b)**a + 1)**(-p - 1)/z >>> cdf(X)(z) Piecewise(((1 + (z/b)**(-a))**(-p), z >= 0), (0, True)) References ========== .. [1] https://en.wikipedia.org/wiki/Dagum_distribution )rr )rrrrrnrnror?s9r?cCst|t|tj|fS)a Create a continuous random variable with an Erlang distribution. Explanation =========== The density of the Erlang distribution is given by .. math:: f(x) := \frac{\lambda^k x^{k-1} e^{-\lambda x}}{(k-1)!} with :math:`x \in [0,\infty]`. Parameters ========== k : Positive integer l : Real number, `\lambda > 0`, the rate Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import Erlang, density, cdf, E, variance >>> from sympy import Symbol, simplify, pprint >>> k = Symbol("k", integer=True, positive=True) >>> l = Symbol("l", positive=True) >>> z = Symbol("z") >>> X = Erlang("x", k, l) >>> D = density(X)(z) >>> pprint(D, use_unicode=False) k k - 1 -l*z l *z *e --------------- Gamma(k) >>> C = cdf(X)(z) >>> pprint(C, use_unicode=False) /lowergamma(k, l*z) |------------------ for z > 0 < Gamma(k) | \ 0 otherwise >>> E(X) k/l >>> simplify(variance(X)) k/l**2 References ========== .. [1] https://en.wikipedia.org/wiki/Erlang_distribution .. [2] http://mathworld.wolfram.com/ErlangDistribution.html )rGammaDistributionrrrrnrnror@ZsCr@c@HeZdZdZee eZeddZddZ ddZ dd Z d d Z d S) ExGaussianDistributionmeanstdratecC t|dkdt|dkddS)Nrz2Standard deviation of ExGaussian must be positive.z$Rate of ExGaussian must be positive.rrrnrnrorvszExGaussianDistribution.checkcCsv|j|j|j}}}|d}t|d|||dd|d}t|||d|td|}|||Sr)rrrrr$r)rrsrrrterm1term2term3rnrnrors ($ zExGaussianDistribution.pdfc Csddlm}|j|j|j}}}|||}||}|tdd||}|td|d||} |t| |ddt| S)Nrcdfrsr) sympy.statsr rrrrWrr) rrsr rrruv GaussianCDF1 GaussianCDF2rnrnrors  "zExGaussianDistribution._cdfcCsV|j|j|j}}}dt||d}tt|||d|dd}||SNrr)rrrrrrrrrrrrrnrnrors$z/ExGaussianDistribution._characteristic_functioncCsN|j|j|j}}}d||d}t|||d|dd}||Sr&)rrrrr(rnrnrors z2ExGaussianDistribution._moment_generating_functionN rrrrr.rrrrvrrrrrnrnrnrors   rcCr)a Create a continuous random variable with an Exponentially modified Gaussian (EMG) distribution. Explanation =========== The density of the exponentially modified Gaussian distribution is given by .. math:: f(x) := \frac{\lambda}{2}e^{\frac{\lambda}{2}(2\mu+\lambda\sigma^2-2x)} \text{erfc}(\frac{\mu + \lambda\sigma^2 - x}{\sqrt{2}\sigma}) with $x > 0$. Note that the expected value is `1/\lambda`. Parameters ========== name : A string giving a name for this distribution mean : A Real number, the mean of Gaussian component std : A positive Real number, :math: `\sigma^2 > 0` the variance of Gaussian component rate : A positive Real number, :math: `\lambda > 0` the rate of Exponential component Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import ExGaussian, density, cdf, E >>> from sympy.stats import variance, skewness >>> from sympy import Symbol, pprint, simplify >>> mean = Symbol("mu") >>> std = Symbol("sigma", positive=True) >>> rate = Symbol("lamda", positive=True) >>> z = Symbol("z") >>> X = ExGaussian("x", mean, std, rate) >>> pprint(density(X)(z), use_unicode=False) / 2 \ lamda*\lamda*sigma + 2*mu - 2*z/ --------------------------------- / ___ / 2 \\ 2 |\/ 2 *\lamda*sigma + mu - z/| lamda*e *erfc|-----------------------------| \ 2*sigma / ---------------------------------------------------------------------------- 2 >>> cdf(X)(z) -(erf(sqrt(2)*(-lamda**2*sigma**2 + lamda*(-mu + z))/(2*lamda*sigma))/2 + 1/2)*exp(lamda**2*sigma**2/2 - lamda*(-mu + z)) + erf(sqrt(2)*(-mu + z)/(2*sigma))/2 + 1/2 >>> E(X) (lamda*mu + 1)/lamda >>> simplify(variance(X)) sigma**2 + lamda**(-2) >>> simplify(skewness(X)) 2/(lamda**2*sigma**2 + 1)**(3/2) References ========== .. [1] https://en.wikipedia.org/wiki/Exponentially_modified_Gaussian_distribution )rr)rrrrrnrnrorAsGrAc@sNeZdZdZedeZeddZddZ ddZ d d Z d d Z d dZ dS)ExponentialDistributionrrcCt|dkddS)NrzRate must be positive.rr+rnrnrorvruzExponentialDistribution.checkcCs|jt|j |Srj)rrrrnrnror"zExponentialDistribution.pdfcCs$ttjt|j ||dkfdS)Nrr)r!rrrrrrnrnror%szExponentialDistribution._cdfcCs|j}||t|Srj)rrrrrrnrnror+sz0ExponentialDistribution._characteristic_functioncCs|j}|||Srjr+r.rnrnror/s z3ExponentialDistribution._moment_generating_functioncCstd| |jSr)rrrrnrnror3sz!ExponentialDistribution._quantileNrrrrr.rrrrvrrrrrrnrnrnror*s   r*cCr)a- Create a continuous random variable with an Exponential distribution. Explanation =========== The density of the exponential distribution is given by .. math:: f(x) := \lambda \exp(-\lambda x) with $x > 0$. Note that the expected value is `1/\lambda`. Parameters ========== rate : A positive Real number, `\lambda > 0`, the rate (or inverse scale/inverse mean) Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import Exponential, density, cdf, E >>> from sympy.stats import variance, std, skewness, quantile >>> from sympy import Symbol >>> l = Symbol("lambda", positive=True) >>> z = Symbol("z") >>> p = Symbol("p") >>> X = Exponential("x", l) >>> density(X)(z) lambda*exp(-lambda*z) >>> cdf(X)(z) Piecewise((1 - exp(-lambda*z), z >= 0), (0, True)) >>> quantile(X)(p) -log(1 - p)/lambda >>> E(X) 1/lambda >>> variance(X) lambda**(-2) >>> skewness(X) 2 >>> X = Exponential('x', 10) >>> density(X)(z) 10*exp(-10*z) >>> E(X) 1/10 >>> std(X) 1/10 References ========== .. [1] https://en.wikipedia.org/wiki/Exponential_distribution .. [2] http://mathworld.wolfram.com/ExponentialDistribution.html )rr*)rrrnrnrorB7sIrBc@s8eZdZdZee eZeddZddZ ddZ dS) ExponentialPowerDistributionmurr cCr)Nrz'Scale parameter alpha must be positive.z&Shape parameter beta must be positive.rr1rnrnrorvrz"ExponentialPowerDistribution.checkcCsP|j|j|j}}}|tt|||| }d|td|}||Sr)r2rr rrr(rrsr2rr rrrnrnrorsz ExponentialPowerDistribution.pdfcCs^|j|j|j}}}td|t||||}dtd|}t||||tjSr) r2rr r)rr(rrrr3rnrnrorsz!ExponentialPowerDistribution._cdfNrrnrnrnror0s   r0cCr)a Create a Continuous Random Variable with Exponential Power distribution. This distribution is known also as Generalized Normal distribution version 1. Explanation =========== The density of the Exponential Power distribution is given by .. math:: f(x) := \frac{\beta}{2\alpha\Gamma(\frac{1}{\beta})} e^{{-(\frac{|x - \mu|}{\alpha})^{\beta}}} with :math:`x \in [ - \infty, \infty ]`. Parameters ========== mu : Real number A location. alpha : Real number,`\alpha > 0` A scale. beta : Real number, `\beta > 0` A shape. Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import ExponentialPower, density, cdf >>> from sympy import Symbol, pprint >>> z = Symbol("z") >>> mu = Symbol("mu") >>> alpha = Symbol("alpha", positive=True) >>> beta = Symbol("beta", positive=True) >>> X = ExponentialPower("x", mu, alpha, beta) >>> pprint(density(X)(z), use_unicode=False) beta /|mu - z|\ -|--------| \ alpha / beta*e --------------------- / 1 \ 2*alpha*Gamma|----| \beta/ >>> cdf(X)(z) 1/2 + lowergamma(1/beta, (Abs(mu - z)/alpha)**beta)*sign(-mu + z)/(2*gamma(1/beta)) References ========== .. [1] https://reference.wolfram.com/language/ref/ExponentialPowerDistribution.html .. [2] https://en.wikipedia.org/wiki/Generalized_normal_distribution#Version_1 )rr0)rr2rr rnrnrorC>rCc@r ) FDistributionDistributiond1d2rcCs,t|dk|jfdt|dk|jfddS)Nrz/Degrees of freedom d1 must be positive integer.z/Degrees of freedom d2 must be positive integer.rr6rnrnrorvs zFDistributionDistribution.checkcCsP|j|j}}t|||||||||||t|d|dSr)r7r8rrrrsr7r8rnrnrors*zFDistributionDistribution.pdfcCr)NzEThe moment generating function for the F-distribution does not exist.rrrnrnrorrz5FDistributionDistribution._moment_generating_functionN) rrrrr.rrrrvrrrnrnrnror5s   r5cCr)a/ Create a continuous random variable with a F distribution. Explanation =========== The density of the F distribution is given by .. math:: f(x) := \frac{\sqrt{\frac{(d_1 x)^{d_1} d_2^{d_2}} {(d_1 x + d_2)^{d_1 + d_2}}}} {x \mathrm{B} \left(\frac{d_1}{2}, \frac{d_2}{2}\right)} with :math:`x > 0`. Parameters ========== d1 : `d_1 > 0`, where `d_1` is the degrees of freedom (`n_1 - 1`) d2 : `d_2 > 0`, where `d_2` is the degrees of freedom (`n_2 - 1`) Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import FDistribution, density >>> from sympy import Symbol, pprint >>> d1 = Symbol("d1", positive=True) >>> d2 = Symbol("d2", positive=True) >>> z = Symbol("z") >>> X = FDistribution("x", d1, d2) >>> D = density(X)(z) >>> pprint(D, use_unicode=False) d2 -- ______________________________ 2 / d1 -d1 - d2 d2 *\/ (d1*z) *(d1*z + d2) -------------------------------------- /d1 d2\ z*B|--, --| \2 2 / References ========== .. [1] https://en.wikipedia.org/wiki/F-distribution .. [2] http://mathworld.wolfram.com/F-Distribution.html )rr5rr7r8rnrnrorD:rDc@s0eZdZdZee eZeddZddZ dS)FisherZDistributionr6cCr)Nrz&Degree of freedom d1 must be positive.z&Degree of freedom d2 must be positive.rr6rnrnrorv;rzFisherZDistribution.checkcCsh|j|j}}d||d||dt|d|dt|||td||||dSr)r7r8rrr9rnrnror@s , zFisherZDistribution.pdfN rrrrr.rrrrvrrnrnrnror<6s    r<cCr)a Create a Continuous Random Variable with an Fisher's Z distribution. Explanation =========== The density of the Fisher's Z distribution is given by .. math:: f(x) := \frac{2d_1^{d_1/2} d_2^{d_2/2}} {\mathrm{B}(d_1/2, d_2/2)} \frac{e^{d_1z}}{\left(d_1e^{2z}+d_2\right)^{\left(d_1+d_2\right)/2}} .. TODO - What is the difference between these degrees of freedom? Parameters ========== d1 : `d_1 > 0` Degree of freedom. d2 : `d_2 > 0` Degree of freedom. Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import FisherZ, density >>> from sympy import Symbol, pprint >>> d1 = Symbol("d1", positive=True) >>> d2 = Symbol("d2", positive=True) >>> z = Symbol("z") >>> X = FisherZ("x", d1, d2) >>> D = density(X)(z) >>> pprint(D, use_unicode=False) d1 d2 d1 d2 - -- - -- -- -- 2 2 2 2 / 2*z \ d1*z 2*d1 *d2 *\d1*e + d2/ *e ----------------------------------------- /d1 d2\ B|--, --| \2 2 / References ========== .. [1] https://en.wikipedia.org/wiki/Fisher%27s_z-distribution .. [2] http://mathworld.wolfram.com/Fishersz-Distribution.html )rr<r:rnrnrorEEs=rEc@s@eZdZdZedeZeddZd ddZ dd Z d d Z d S)FrechetDistributionrsmrcCr)Nrz'Shape parameter alpha must be positive.#Scale parameter s must be positive.rr?rnrnrorvrzFrechetDistribution.checkrcCs*ttt|||f\}}}t||||Srj)rzr{rr r)rrr@rArnrnrorszFrechetDistribution.__new__cCsJ|j|j|j}}}|||||d|t||||  S)Nr')rr@rArrrsrr@rArnrnrorrzFrechetDistribution.pdfcCsB|j|j|j}}}tt||||  ||kftjdfSNT)rr@rAr!rrrrCrnrnrors zFrechetDistribution._cdfNrr) rrrrr.rrrrvrrrrnrnrnror>s    r>cCr)a Create a continuous random variable with a Frechet distribution. Explanation =========== The density of the Frechet distribution is given by .. math:: f(x) := \frac{\alpha}{s} \left(\frac{x-m}{s}\right)^{-1-\alpha} e^{-(\frac{x-m}{s})^{-\alpha}} with :math:`x \geq m`. Parameters ========== a : Real number, :math:`a \in \left(0, \infty\right)` the shape s : Real number, :math:`s \in \left(0, \infty\right)` the scale m : Real number, :math:`m \in \left(-\infty, \infty\right)` the minimum Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import Frechet, density, cdf >>> from sympy import Symbol >>> a = Symbol("a", positive=True) >>> s = Symbol("s", positive=True) >>> m = Symbol("m", real=True) >>> z = Symbol("z") >>> X = Frechet("x", a, s, m) >>> density(X)(z) a*((-m + z)/s)**(-a - 1)*exp(-1/((-m + z)/s)**a)/s >>> cdf(X)(z) Piecewise((exp(-1/((-m + z)/s)**a), m <= z), (0, True)) References ========== .. [1] https://en.wikipedia.org/wiki/Fr%C3%A9chet_distribution )rr>)rrr@rArnrnrorFs5rFc@FeZdZdZedeZeddZddZ ddZ d d Z d d Z d S)rrthetarcCr)Nrk must be positivezTheta must be positiverrGrnrnrorvrzGammaDistribution.checkcCs8|j|j}}||dt| |t|||Sr)rrHrr(rrsrrHrnrnror*zGammaDistribution.pdfcCs<|j|j}}tt|t||t||dkftjdfSr)rrHr!r)rr(rrJrnrnrors  zGammaDistribution._cdfcCsd|jt||j Sr)rHrrrrnrnrorrz*GammaDistribution._characteristic_functioncCsd|j||j Sr)rHrrrnrnrorr-z-GammaDistribution._moment_generating_functionNr)rnrnrnrors   rcCr)a Create a continuous random variable with a Gamma distribution. Explanation =========== The density of the Gamma distribution is given by .. math:: f(x) := \frac{1}{\Gamma(k) \theta^k} x^{k - 1} e^{-\frac{x}{\theta}} with :math:`x \in [0,1]`. Parameters ========== k : Real number, `k > 0`, a shape theta : Real number, `\theta > 0`, a scale Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import Gamma, density, cdf, E, variance >>> from sympy import Symbol, pprint, simplify >>> k = Symbol("k", positive=True) >>> theta = Symbol("theta", positive=True) >>> z = Symbol("z") >>> X = Gamma("x", k, theta) >>> D = density(X)(z) >>> pprint(D, use_unicode=False) -z ----- -k k - 1 theta theta *z *e --------------------- Gamma(k) >>> C = cdf(X, meijerg=True)(z) >>> pprint(C, use_unicode=False) / / z \ |k*lowergamma|k, -----| | \ theta/ <---------------------- for z >= 0 | Gamma(k + 1) | \ 0 otherwise >>> E(X) k*theta >>> V = simplify(variance(X)) >>> pprint(V, use_unicode=False) 2 k*theta References ========== .. [1] https://en.wikipedia.org/wiki/Gamma_distribution .. [2] http://mathworld.wolfram.com/GammaDistribution.html )rr)rrrHrnrnrorGsIrGc@rF)GammaInverseDistributionrrcCr)Nrzalpha must be positivezbeta must be positiverrrnrnrorvHrzGammaInverseDistribution.checkcCs:|j|j}}||t||| dt| |Sr)rrr(rrrnrnrorM,zGammaInverseDistribution.pdfcCs8|j|j}}tt|||t||dkftjdfSr)rrr!r*r(rrrrnrnrorQszGammaInverseDistribution._cdfcCsJ|j|j}}dt |||dt|tdt||t|S)Nr)rrrr rr(rrrrrnrnrorVs<z1GammaInverseDistribution._characteristic_functioncCr)NzQThe moment generating function for the gamma inverse distribution does not exist.rrrnrnrorZrz4GammaInverseDistribution._moment_generating_functionNr)rnrnrnrorLCs   rLcCr)aY Create a continuous random variable with an inverse Gamma distribution. Explanation =========== The density of the inverse Gamma distribution is given by .. math:: f(x) := \frac{\beta^\alpha}{\Gamma(\alpha)} x^{-\alpha - 1} \exp\left(\frac{-\beta}{x}\right) with :math:`x > 0`. Parameters ========== a : Real number, `a > 0`, a shape b : Real number, `b > 0`, a scale Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import GammaInverse, density, cdf >>> from sympy import Symbol, pprint >>> a = Symbol("a", positive=True) >>> b = Symbol("b", positive=True) >>> z = Symbol("z") >>> X = GammaInverse("x", a, b) >>> D = density(X)(z) >>> pprint(D, use_unicode=False) -b --- a -a - 1 z b *z *e --------------- Gamma(a) >>> cdf(X)(z) Piecewise((uppergamma(a, b/z)/gamma(a), z > 0), (0, True)) References ========== .. [1] https://en.wikipedia.org/wiki/Inverse-gamma_distribution )rrLrrnrnrorH^r;rHc@r) GumbelDistributionr r2minimumcCr,)Nrz&Scale parameter beta must be positive.rrQrnrnrorvruzGumbelDistribution.checkcCsf|j|j}}|||}d|t| t| }d|t|t|}t||jf||j fSrr r2rr!rR)rrsr r2zf_maxf_minrnrnrors  zGumbelDistribution.pdfcCsT|j|j}}|||}tt|  }dtt| }t||jf||j fSrrS)rrsr r2rTF_maxF_minrnrnrors  zGumbelDistribution._cdfcCshtdt|j|tt|j|}tdt|j|tt|j|}t||jf||j fSr)r(rr rr2r!rR)rrcf_maxcf_minrnrnrors((z+GumbelDistribution._characteristic_functioncCsXtd|j|t|j|}td|j|t|j|}t||jf||j fSr)r(r rr2r!rR)rrmgf_maxmgf_minrnrnrors  z.GumbelDistribution._moment_generating_functionNr)rnrnrnrorPs   rPFcCr)a Create a Continuous Random Variable with Gumbel distribution. Explanation =========== The density of the Gumbel distribution is given by For Maximum .. math:: f(x) := \dfrac{1}{\beta} \exp \left( -\dfrac{x-\mu}{\beta} - \exp \left( -\dfrac{x - \mu}{\beta} \right) \right) with :math:`x \in [ - \infty, \infty ]`. For Minimum .. math:: f(x) := \frac{e^{- e^{\frac{- \mu + x}{\beta}} + \frac{- \mu + x}{\beta}}}{\beta} with :math:`x \in [ - \infty, \infty ]`. Parameters ========== mu : Real number, `\mu`, a location beta : Real number, `\beta > 0`, a scale minimum : Boolean, by default ``False``, set to ``True`` for enabling minimum distribution Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import Gumbel, density, cdf >>> from sympy import Symbol >>> x = Symbol("x") >>> mu = Symbol("mu") >>> beta = Symbol("beta", positive=True) >>> X = Gumbel("x", beta, mu) >>> density(X)(x) exp(-exp(-(-mu + x)/beta) - (-mu + x)/beta)/beta >>> cdf(X)(x) exp(-exp(-(-mu + x)/beta)) References ========== .. [1] http://mathworld.wolfram.com/GumbelDistribution.html .. [2] https://en.wikipedia.org/wiki/Gumbel_distribution .. [3] http://www.mathwave.com/help/easyfit/html/analyses/distributions/gumbel_max.html .. [4] http://www.mathwave.com/help/easyfit/html/analyses/distributions/gumbel_min.html )rrP)rr r2rRrnrnrorJs;rJc@>eZdZdZedeZeddZddZ ddZ d d Z d S) GompertzDistributionretarcCrNrb must be positivezeta must be positiverr_rnrnrorvrzGompertzDistribution.checkcCs@|j|j}}||t||t|t| t||Srjr`rrrrsr`rrnrnror s2zGompertzDistribution.pdfcCs0|j|j}}dt|t| t||Srrcrdrnrnror"zGompertzDistribution._cdfcCs(|j|j}}|t|t|||Srj)r`rrr')rrr`rrnrnrorsz0GompertzDistribution._moment_generating_functionN rrrrr.rrrrvrrrrnrnrnror^s   r^cCr)a_ Create a Continuous Random Variable with Gompertz distribution. Explanation =========== The density of the Gompertz distribution is given by .. math:: f(x) := b \eta e^{b x} e^{\eta} \exp \left(-\eta e^{bx} \right) with :math:`x \in [0, \infty)`. Parameters ========== b : Real number, `b > 0`, a scale eta : Real number, `\eta > 0`, a shape Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import Gompertz, density >>> from sympy import Symbol >>> b = Symbol("b", positive=True) >>> eta = Symbol("eta", positive=True) >>> z = Symbol("z") >>> X = Gompertz("x", b, eta) >>> density(X)(z) b*eta*exp(eta)*exp(b*z)*exp(-eta*exp(b*z)) References ========== .. [1] https://en.wikipedia.org/wiki/Gompertz_distribution )rr^rrr`rnrnrorI.rIc@r ) KumaraswamyDistributionrrcCr)Nrza must be positiverbrrrnrnrorvOrzKumaraswamyDistribution.checkcCs6|j|j}}||||dd|||dSrrrrnrnrorTrzKumaraswamyDistribution.pdfcCsF|j|j}}ttj|tjkfdd||||tjkftjdfSNrT)rrr!rrrrrnrnrorXs zKumaraswamyDistribution._cdfNrrnrnrnroriJ   ricCr)a Create a Continuous Random Variable with a Kumaraswamy distribution. Explanation =========== The density of the Kumaraswamy distribution is given by .. math:: f(x) := a b x^{a-1} (1-x^a)^{b-1} with :math:`x \in [0,1]`. Parameters ========== a : Real number, `a > 0`, a shape b : Real number, `b > 0`, a shape Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import Kumaraswamy, density, cdf >>> from sympy import Symbol, pprint >>> a = Symbol("a", positive=True) >>> b = Symbol("b", positive=True) >>> z = Symbol("z") >>> X = Kumaraswamy("x", a, b) >>> D = density(X)(z) >>> pprint(D, use_unicode=False) b - 1 a - 1 / a\ a*b*z *\1 - z / >>> cdf(X)(z) Piecewise((0, z < 0), (1 - (1 - z**a)**b, z <= 1), (1, True)) References ========== .. [1] https://en.wikipedia.org/wiki/Kumaraswamy_distribution )rrirrnrnrorK_s5rKc@r) LaplaceDistributionr2rcCr)Nrrz$Location parameter mu should be realrrmrnrnrorvrzLaplaceDistribution.checkcCs0|j|j}}dd|tt|| |Sr)r2rrrrrsr2rrnrnrorrezLaplaceDistribution.pdfcCsT|j|j}}ttjt|||||kftjtjt|| |||kfSrj)r2rr!rrrrrnrnrnrors $zLaplaceDistribution._cdfcCs(t|jt|d|jd|dSr)rr2rrrrnrnrors(z,LaplaceDistribution._characteristic_functioncCs$t|j|d|jd|dSr)rr2rrrnrnrors$z/LaplaceDistribution._moment_generating_functionNr)rnrnrnrorls   rlcCsDt|ttfrt|ttfrddlm}||||St|t||fS)a Create a continuous random variable with a Laplace distribution. Explanation =========== The density of the Laplace distribution is given by .. math:: f(x) := \frac{1}{2 b} \exp \left(-\frac{|x-\mu|}b \right) Parameters ========== mu : Real number or a list/matrix, the location (mean) or the location vector b : Real number or a positive definite matrix, representing a scale or the covariance matrix. Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import Laplace, density, cdf >>> from sympy import Symbol, pprint >>> mu = Symbol("mu") >>> b = Symbol("b", positive=True) >>> z = Symbol("z") >>> X = Laplace("x", mu, b) >>> density(X)(z) exp(-Abs(mu - z)/b)/(2*b) >>> cdf(X)(z) Piecewise((exp((-mu + z)/b)/2, mu > z), (1 - exp((mu - z)/b)/2, True)) >>> L = Laplace('L', [1, 2], [[1, 0], [0, 1]]) >>> pprint(density(L)(1, 2), use_unicode=False) 5 / ____\ e *besselk\0, \/ 35 / --------------------- pi References ========== .. [1] https://en.wikipedia.org/wiki/Laplace_distribution .. [2] http://mathworld.wolfram.com/LaplaceDistribution.html r)MultivariateLaplace) isinstancerzr/sympy.stats.joint_rv_typesrorrl)rr2rrornrnrorLs :   rLc@HeZdZdZeddZeddZddZdd Z d d Z d d Z dS)LevyDistributionr2ccCrrj)r.r2rrrnrnrorrzLevyDistribution.setcCr)Nrz$c (scale parameter) must be positivez$mu (location paramater) must be realrrtrnrnrorv rzLevyDistribution.checkcCsH|j|j}}t|dtt| d||||tjtjSr)r2rurrrrrrrrsr2rurnrnror s:zLevyDistribution.pdfcCs&|j|j}}tt|d||Sr)r2rur$rrvrnrnror zLevyDistribution._cdfcCs2|j|j}}tt||tdt||S)N)r2rurrr)rrr2rurnrnror $z)LevyDistribution._characteristic_functioncCr)NzCThe moment generating function of Levy distribution does not exist.rrrnrnror rz,LevyDistribution._moment_generating_functionN rrrrrrrrvrrrrrnrnrnrorss   rscCr)a Create a continuous random variable with a Levy distribution. The density of the Levy distribution is given by .. math:: f(x) := \sqrt(\frac{c}{2 \pi}) \frac{\exp -\frac{c}{2 (x - \mu)}}{(x - \mu)^{3/2}} Parameters ========== mu : Real number The location parameter. c : Real number, `c > 0` A scale parameter. Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import Levy, density, cdf >>> from sympy import Symbol >>> mu = Symbol("mu", real=True) >>> c = Symbol("c", positive=True) >>> z = Symbol("z") >>> X = Levy("x", mu, c) >>> density(X)(z) sqrt(2)*sqrt(c)*exp(-c/(-2*mu + 2*z))/(2*sqrt(pi)*(-mu + z)**(3/2)) >>> cdf(X)(z) erfc(sqrt(c)*sqrt(1/(-2*mu + 2*z))) References ========== .. [1] https://en.wikipedia.org/wiki/L%C3%A9vy_distribution .. [2] http://mathworld.wolfram.com/LevyDistribution.html )rrs)rr2rurnrnrorM rhrMc@sHeZdZdZedeZeddZ ddZ ddZ d d Z d d Z d S)LogCauchyDistributionr2rrcCs&t|dkdkdt|jdkddS)NrFrrrr|rnrnrorvN szLogCauchyDistribution.checkcCs6|j|j}}d|t|t||d|dSr)r2rrrrrsr2rrnrnrorS rzLogCauchyDistribution.pdfcCs0|j|j}}dttt|||tjSr)r2rrrrrrr}rnrnrorW rezLogCauchyDistribution._cdfcCr)NzKThe characteristic function for the Log-Cauchy distribution does not exist.rrrnrnror[ rz.LogCauchyDistribution._characteristic_functioncCr)NzNThe moment generating function for the Log-Cauchy distribution does not exist.rrrnrnror_ rz1LogCauchyDistribution._moment_generating_functionN)rrrrr.openrrrrvrrrrrnrnrnror{I s   r{cCr)a3 Create a continuous random variable with a Log-Cauchy distribution. The density of the Log-Cauchy distribution is given by .. math:: f(x) := \frac{1}{\pi x} \frac{\sigma}{(log(x)-\mu^2) + \sigma^2} Parameters ========== mu : Real number, the location sigma : Real number, `\sigma > 0`, a scale Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import LogCauchy, density, cdf >>> from sympy import Symbol, S >>> mu = 2 >>> sigma = S.One / 5 >>> z = Symbol("z") >>> X = LogCauchy("x", mu, sigma) >>> density(X)(z) 1/(5*pi*z*((log(z) - 2)**2 + 1/25)) >>> cdf(X)(z) atan(5*log(z) - 10)/pi + 1/2 References ========== .. [1] https://en.wikipedia.org/wiki/Log-Cauchy_distribution )rr{rr2rrnrnrorNc s,rNc@sPeZdZdZee eZeddZddZ ddZ dd Z d d Z d d Z dS)LogisticDistributionr2r@cCt|dkddS)NrrBrrrnrnrorv ruzLogisticDistribution.checkcCs>|j|j}}t|| ||dt|| |dSr)r2r@rrrsr2r@rnrnror 0zLogisticDistribution.pdfcCs*|j|j}}tjdt|| |Sr)r2r@rrrrrnrnror rzLogisticDistribution._cdfcCsHttt||jt|j|tt|j|t|dftj dfSr) r!rrr2rr@rrrrrrnrnror sHz-LogisticDistribution._characteristic_functioncCs,t|j|td|j|d|j|Sr)rr2rr@rrnrnror s,z0LogisticDistribution._moment_generating_functioncCs"|j|jttj tj|Srj)r2r@rrrrrnrnror s"zLogisticDistribution._quantileNr/rnrnrnror s   rcCr)a Create a continuous random variable with a logistic distribution. Explanation =========== The density of the logistic distribution is given by .. math:: f(x) := \frac{e^{-(x-\mu)/s}} {s\left(1+e^{-(x-\mu)/s}\right)^2} Parameters ========== mu : Real number, the location (mean) s : Real number, `s > 0`, a scale Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import Logistic, density, cdf >>> from sympy import Symbol >>> mu = Symbol("mu", real=True) >>> s = Symbol("s", positive=True) >>> z = Symbol("z") >>> X = Logistic("x", mu, s) >>> density(X)(z) exp((mu - z)/s)/(s*(exp((mu - z)/s) + 1)**2) >>> cdf(X)(z) 1/(exp((mu - z)/s) + 1) References ========== .. [1] https://en.wikipedia.org/wiki/Logistic_distribution .. [2] http://mathworld.wolfram.com/LogisticDistribution.html )rrrr2r@rnrnrorO rrOc@rF)LogLogisticDistributionrrcCr)Nrz'Scale parameter Alpha must be positive.rrrrnrnrorv rzLogLogisticDistribution.checkcCs:|j|j}}|||||dd|||dSrrrrnrnror rMzLogLogisticDistribution.pdfcCs$|j|j}}dd||| Srrrrnrnror szLogLogisticDistribution._cdfcCs&|j|j}}||d|d|Srr)rrrrrnrnror rwz!LogLogisticDistribution._quantilecKs6|j\}}ttj|dkft||tt|dfSrj)rr!rNaNrr)rexprvarrrrrnrnro expectation s ,z#LogLogisticDistribution.expectationN) rrrrr.rrrrvrrrrrnrnrnror s   rcCr)a Create a continuous random variable with a log-logistic distribution. The distribution is unimodal when ``beta > 1``. Explanation =========== The density of the log-logistic distribution is given by .. math:: f(x) := \frac{(\frac{\beta}{\alpha})(\frac{x}{\alpha})^{\beta - 1}} {(1 + (\frac{x}{\alpha})^{\beta})^2} Parameters ========== alpha : Real number, `\alpha > 0`, scale parameter and median of distribution beta : Real number, `\beta > 0`, a shape parameter Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import LogLogistic, density, cdf, quantile >>> from sympy import Symbol, pprint >>> alpha = Symbol("alpha", positive=True) >>> beta = Symbol("beta", positive=True) >>> p = Symbol("p") >>> z = Symbol("z", positive=True) >>> X = LogLogistic("x", alpha, beta) >>> D = density(X)(z) >>> pprint(D, use_unicode=False) beta - 1 / z \ beta*|-----| \alpha/ ------------------------ 2 / beta \ |/ z \ | alpha*||-----| + 1| \\alpha/ / >>> cdf(X)(z) 1/(1 + (z/alpha)**(-beta)) >>> quantile(X)(p) alpha*(p/(1 - p))**(1/beta) References ========== .. [1] https://en.wikipedia.org/wiki/Log-logistic_distribution )rrrrnrnrorP s@rPc@s@eZdZdZeddZeddZddZ dd Z d d Z d S) LogitNormalDistributionrrrcCs4t|djduo |ddkdt|jduddS)NrFrz+Squared scale parameter s must be positive.zLocation parameter must be realrrrnrnrorvJ s zLogitNormalDistribution.checkcCst|d|Srrrrnrnro_logitO szLogitNormalDistribution._logitcCs\|j|j}}t|||d d|dtjtdt|dd|d|Sr)r2r@rrrrrrrrnrnrorR sNzLogitNormalDistribution.pdfcCs>|j|j}}tjddt|||td|dSr)r2r@rrr#rrrrnrnrorV rzLogitNormalDistribution._cdfN) rrrrr.r~rrrvrrrrnrnrnrorF s   rcCr)aE Create a continuous random variable with a Logit-Normal distribution. The density of the logistic distribution is given by .. math:: f(x) := \frac{1}{s \sqrt{2 \pi}} \frac{1}{x(1 - x)} e^{- \frac{(logit(x) - \mu)^2}{s^2}} where logit(x) = \log(\frac{x}{1 - x}) Parameters ========== mu : Real number, the location (mean) s : Real number, `s > 0`, a scale Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import LogitNormal, density, cdf >>> from sympy import Symbol,pprint >>> mu = Symbol("mu", real=True) >>> s = Symbol("s", positive=True) >>> z = Symbol("z") >>> X = LogitNormal("x",mu,s) >>> D = density(X)(z) >>> pprint(D, use_unicode=False) 2 / / z \\ -|-mu + log|-----|| \ \1 - z// --------------------- 2 ___ 2*s \/ 2 *e ---------------------------- ____ 2*\/ pi *s*z*(1 - z) >>> density(X)(z) sqrt(2)*exp(-(-mu + log(z/(1 - z)))**2/(2*s**2))/(2*sqrt(pi)*s*z*(1 - z)) >>> cdf(X)(z) erf(sqrt(2)*(-mu + log(z/(1 - z)))/(2*s))/2 + 1/2 References ========== .. [1] https://en.wikipedia.org/wiki/Logit-normal_distribution )rrrrnrnrorQ[ s;rQc@r]) LogNormalDistributionrrrcCr)NrzParameter std must be positive.rrrnrnrorv ruzLogNormalDistribution.checkcCsD|j|j}}tt||d d|d|tdt|Sr)rrrrrrrrsrrrnrnror s6zLogNormalDistribution.pdfcCsJ|j|j}}ttjtjtt||td||dkftjdfSNrrT) rrr!rrr#rrrrrnrnror s .zLogNormalDistribution._cdfcCr)NzIMoment generating function of the log-normal distribution is not defined.rrrnrnror rz1LogNormalDistribution._moment_generating_functionNrfrnrnrnror s   rcCr)a4 Create a continuous random variable with a log-normal distribution. Explanation =========== The density of the log-normal distribution is given by .. math:: f(x) := \frac{1}{x\sqrt{2\pi\sigma^2}} e^{-\frac{\left(\ln x-\mu\right)^2}{2\sigma^2}} with :math:`x \geq 0`. Parameters ========== mu : Real number The log-scale. sigma : Real number A shape. ($\sigma^2 > 0$) Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import LogNormal, density >>> from sympy import Symbol, pprint >>> mu = Symbol("mu", real=True) >>> sigma = Symbol("sigma", positive=True) >>> z = Symbol("z") >>> X = LogNormal("x", mu, sigma) >>> D = density(X)(z) >>> pprint(D, use_unicode=False) 2 -(-mu + log(z)) ----------------- 2 ___ 2*sigma \/ 2 *e ------------------------ ____ 2*\/ pi *sigma*z >>> X = LogNormal('x', 0, 1) # Mean 0, standard deviation 1 >>> density(X)(z) sqrt(2)*exp(-log(z)**2/2)/(2*sqrt(pi)*z) References ========== .. [1] https://en.wikipedia.org/wiki/Lognormal .. [2] http://mathworld.wolfram.com/LogNormalDistribution.html )rr)rrrrnrnrorR BrRc@.eZdZdZedeZeddZddZ dS)LomaxDistributionrrrcCs4t|jdt|jdt|jdt|jddS)NzShape parameter should be real.zScale parameter should be real.z#Shape parameter should be positive.z#Scale parameter should be positive.r2rrrrnrnrorv s   zLomaxDistribution.checkcCs.|j|j}}||tj||| dSr)rrrr)rrslambarrnrnror  zLomaxDistribution.pdfNr=rnrnrnror s    rcCr)a Create a continuous random variable with a Lomax distribution. Explanation =========== The density of the Lomax distribution is given by .. math:: f(x) := \frac{\alpha}{\lambda}\left[1+\frac{x}{\lambda}\right]^{-(\alpha+1)} Parameters ========== alpha : Real Number, `\alpha > 0` Shape parameter lamda : Real Number, `\lambda > 0` Scale parameter Examples ======== >>> from sympy.stats import Lomax, density, cdf, E >>> from sympy import symbols >>> a, l = symbols('a, l', positive=True) >>> X = Lomax('X', a, l) >>> x = symbols('x') >>> density(X)(x) a*(1 + x/l)**(-a - 1)/l >>> cdf(X)(x) Piecewise((1 - 1/(1 + x/l)**a, x >= 0), (0, True)) >>> a = 2 >>> X = Lomax('X', a, l) >>> E(X) l Returns ======= RandomSymbol References ========== .. [1] https://en.wikipedia.org/wiki/Lomax_distribution )rr)rrrrnrnrorS 0rSc@r ) MaxwellDistributionrrcCr,)NrzParameter a must be positive.rrrnrnrorvE ruzMaxwellDistribution.checkcCs<|j}tdt|dt|d d|d|dSNrr)rrrrrrsrrnrnrorI s6zMaxwellDistribution.pdfcCsP|j}ttd|d|td|t|d d|dtt|Sr)rr#rrrrrnrnrorM JzMaxwellDistribution._cdfNrrnrnrnror@ s   rcCr)a Create a continuous random variable with a Maxwell distribution. Explanation =========== The density of the Maxwell distribution is given by .. math:: f(x) := \sqrt{\frac{2}{\pi}} \frac{x^2 e^{-x^2/(2a^2)}}{a^3} with :math:`x \geq 0`. .. TODO - what does the parameter mean? Parameters ========== a : Real number, `a > 0` Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import Maxwell, density, E, variance >>> from sympy import Symbol, simplify >>> a = Symbol("a", positive=True) >>> z = Symbol("z") >>> X = Maxwell("x", a) >>> density(X)(z) sqrt(2)*z**2*exp(-z**2/(2*a**2))/(sqrt(pi)*a**3) >>> E(X) 2*sqrt(2)*a/sqrt(pi) >>> simplify(variance(X)) a**2*(-8 + 3*pi)/pi References ========== .. [1] https://en.wikipedia.org/wiki/Maxwell_distribution .. [2] http://mathworld.wolfram.com/MaxwellDistribution.html )rr)rrrnrnrorTQ s6rTc@s4eZdZdZeddZddZddZdd Zd S) MoyalDistributionr|cCs$t|jdt|jo |dkddS)Nrrz1Scale parameter must be real and positive.rr|rnrnrorv s zMoyalDistribution.checkcCsN|j|j}}tt|| |||| d}tdt|}||Sr)r2rrrr)rrsr2rrrrnrnror s(zMoyalDistribution.pdfcCsZ|j|j}}tt||}dt ||ttddt||}||ttSr)r2rrrr(rrrrrr2rrrrnrnror s,z*MoyalDistribution._characteristic_functioncCsP|j|j}}t||}dd||ttdd||}||ttS)Nrr'r)r2rrr(rrrrrnrnror s &z-MoyalDistribution._moment_generating_functionN) rrrrrrvrrrrnrnrnror s  rcCr)aw Create a continuous random variable with a Moyal distribution. Explanation =========== The density of the Moyal distribution is given by .. math:: f(x) := \frac{\exp-\frac{1}{2}\exp-\frac{x-\mu}{\sigma}-\frac{x-\mu}{2\sigma}}{\sqrt{2\pi}\sigma} with :math:`x \in \mathbb{R}`. Parameters ========== mu : Real number Location parameter sigma : Real positive number Scale parameter Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import Moyal, density, cdf >>> from sympy import Symbol, simplify >>> mu = Symbol("mu", real=True) >>> sigma = Symbol("sigma", positive=True, real=True) >>> z = Symbol("z") >>> X = Moyal("x", mu, sigma) >>> density(X)(z) sqrt(2)*exp(-exp((mu - z)/sigma)/2 - (-mu + z)/(2*sigma))/(2*sqrt(pi)*sigma) >>> simplify(cdf(X)(z)) 1 - erf(sqrt(2)*exp((mu - z)/(2*sigma))/2) References ========== .. [1] https://reference.wolfram.com/language/ref/MoyalDistribution.html .. [2] http://www.stat.rice.edu/~dobelman/textfiles/DistributionsHandbook.pdf )rrrrnrnrorU rrUc@r ) NakagamiDistributionr2omegarcCs"t|tjkdt|dkddS)Nz5Shape parameter mu must be greater than equal to 1/2.rz(Spread parameter omega must be positive.)r2rrrrnrnrorv szNakagamiDistribution.checkcCsP|j|j}}d||t||||d|dt| ||dSr)r2rr(rrrsr2rrnrnror sBzNakagamiDistribution.pdfcCs@|j|j}}tt||||dt||dkftjdfSr)r2rr!r)r(rrrrnrnror s $zNakagamiDistribution._cdfNrrnrnrnror rkrcCr)aF Create a continuous random variable with a Nakagami distribution. Explanation =========== The density of the Nakagami distribution is given by .. math:: f(x) := \frac{2\mu^\mu}{\Gamma(\mu)\omega^\mu} x^{2\mu-1} \exp\left(-\frac{\mu}{\omega}x^2 \right) with :math:`x > 0`. Parameters ========== mu : Real number, `\mu \geq \frac{1}{2}`, a shape omega : Real number, `\omega > 0`, the spread Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import Nakagami, density, E, variance, cdf >>> from sympy import Symbol, simplify, pprint >>> mu = Symbol("mu", positive=True) >>> omega = Symbol("omega", positive=True) >>> z = Symbol("z") >>> X = Nakagami("x", mu, omega) >>> D = density(X)(z) >>> pprint(D, use_unicode=False) 2 -mu*z ------- mu -mu 2*mu - 1 omega 2*mu *omega *z *e ---------------------------------- Gamma(mu) >>> simplify(E(X)) sqrt(mu)*sqrt(omega)*gamma(mu + 1/2)/gamma(mu + 1) >>> V = simplify(variance(X)) >>> pprint(V, use_unicode=False) 2 omega*Gamma (mu + 1/2) omega - ----------------------- Gamma(mu)*Gamma(mu + 1) >>> cdf(X)(z) Piecewise((lowergamma(mu, mu*z**2/omega)/gamma(mu), z > 0), (0, True)) References ========== .. [1] https://en.wikipedia.org/wiki/Nakagami_distribution )rr)rr2rrnrnrorV sFrVc@r)NormalDistributionrcCr)Nrz#Standard deviation must be positiverrrnrnrorv@ ruzNormalDistribution.checkcCs4t||jd d|jdtdt|jSr)rrrrrrrnrnrorD s4zNormalDistribution.pdfcCs6|j|j}}ttd| |d|dtjSr)rrr#rrrrrnrnrorG rzNormalDistribution._cdfcCs2|j|j}}tt|||d|ddSr)rrrrrrrrrnrnrorK ryz+NormalDistribution._characteristic_functioncCs.|j|j}}t|||d|ddSr)rrrrrnrnrorO rz.NormalDistribution._moment_generating_functioncCs.|j|j}}||tdtd|dSr)rrrr&)rrrrrnrnrorS rzNormalDistribution._quantileNrrnrnrnror= s  rcCsTt|tst|ddrt|tst|ddr"ddlm}||||St|t||fS)a Create a continuous random variable with a Normal distribution. Explanation =========== The density of the Normal distribution is given by .. math:: f(x) := \frac{1}{\sigma\sqrt{2\pi}} e^{ -\frac{(x-\mu)^2}{2\sigma^2} } Parameters ========== mu : Real number or a list representing the mean or the mean vector sigma : Real number or a positive definite square matrix, :math:`\sigma^2 > 0`, the variance Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import Normal, density, E, std, cdf, skewness, quantile, marginal_distribution >>> from sympy import Symbol, simplify, pprint >>> mu = Symbol("mu") >>> sigma = Symbol("sigma", positive=True) >>> z = Symbol("z") >>> y = Symbol("y") >>> p = Symbol("p") >>> X = Normal("x", mu, sigma) >>> density(X)(z) sqrt(2)*exp(-(-mu + z)**2/(2*sigma**2))/(2*sqrt(pi)*sigma) >>> C = simplify(cdf(X))(z) # it needs a little more help... >>> pprint(C, use_unicode=False) / ___ \ |\/ 2 *(-mu + z)| erf|---------------| \ 2*sigma / 1 -------------------- + - 2 2 >>> quantile(X)(p) mu + sqrt(2)*sigma*erfinv(2*p - 1) >>> simplify(skewness(X)) 0 >>> X = Normal("x", 0, 1) # Mean 0, standard deviation 1 >>> density(X)(z) sqrt(2)*exp(-z**2/2)/(2*sqrt(pi)) >>> E(2*X + 1) 1 >>> simplify(std(2*X + 1)) 2 >>> m = Normal('X', [1, 2], [[2, 1], [1, 2]]) >>> pprint(density(m)(y, z), use_unicode=False) 2 2 y y*z z - -- + --- - -- + z - 1 ___ 3 3 3 \/ 3 *e ------------------------------ 6*pi >>> marginal_distribution(m, m[0])(1) 1/(2*sqrt(pi)) References ========== .. [1] https://en.wikipedia.org/wiki/Normal_distribution .. [2] http://mathworld.wolfram.com/NormalDistributionFunction.html is_MatrixFr)MultivariateNormal)rprzgetattrrqrrr)rrrrrnrnrorWX sW   rWc@rr)GaussianInverseDistributionrshapecCs tdtSNr)r.rrrnrnror rzGaussianInverseDistribution.setcCs t|dkdt|dkddS)Nrz Shape parameter must be positivezMean must be positiverrrnrnrorv rz!GaussianInverseDistribution.checkcCsL|j|j}}t| ||dd||dt|dt|dSr)rrrrrrrnrnror s>zGaussianInverseDistribution.pdfcCsddlm}|j|j}}|tddd}|t||||tj}td|||t|| ||tj}||S)Nrrrsrr) r!r rrrWrrrr)rrsr r2r@ stdNormalcdf first_term second_termrnrnror s 0z GaussianInverseDistribution._cdfcCs>|j|j}}t||dtdd|dt||Sr)rrrrrrrr2r@rnrnror rz4GaussianInverseDistribution._characteristic_functioncCs:|j|j}}t||dtdd|d||Sr)rrrrrrnrnror rMz7GaussianInverseDistribution._moment_generating_functionNrzrnrnrnror s   rcCr)a Create a continuous random variable with an Inverse Gaussian distribution. Inverse Gaussian distribution is also known as Wald distribution. Explanation =========== The density of the Inverse Gaussian distribution is given by .. math:: f(x) := \sqrt{\frac{\lambda}{2\pi x^3}} e^{-\frac{\lambda(x-\mu)^2}{2x\mu^2}} Parameters ========== mu : Positive number representing the mean. lambda : Positive number representing the shape parameter. Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import GaussianInverse, density, E, std, skewness >>> from sympy import Symbol, pprint >>> mu = Symbol("mu", positive=True) >>> lamda = Symbol("lambda", positive=True) >>> z = Symbol("z", positive=True) >>> X = GaussianInverse("x", mu, lamda) >>> D = density(X)(z) >>> pprint(D, use_unicode=False) 2 -lambda*(-mu + z) ------------------- 2 ___ ________ 2*mu *z \/ 2 *\/ lambda *e ------------------------------------- ____ 3/2 2*\/ pi *z >>> E(X) mu >>> std(X).expand() mu**(3/2)/sqrt(lambda) >>> skewness(X).expand() 3*sqrt(mu)/sqrt(lambda) References ========== .. [1] https://en.wikipedia.org/wiki/Inverse_Gaussian_distribution .. [2] http://mathworld.wolfram.com/InverseGaussianDistribution.html )rr)rrrrnrnrorX rrXc@rr)ParetoDistributionxmrcCrrj)r.rrrrnrnror* rzParetoDistribution.setcCr)NrzXm must be positiveAlpha must be positiverrrnrnrorv. rzParetoDistribution.checkcCs&|j|j}}|||||dSrrrrsrrrnrnror3 rwzParetoDistribution.pdfcCs2|j|j}}ttj||||||kfdS)Nr)rrr!rrrrnrnror7 s zParetoDistribution._cdfcCs2|j|j}}|| ||t| | |Srj)rrr*rrrrrnrnror> ryz.ParetoDistribution._moment_generating_functioncCs:|j|j}}|t |||t| t ||Srj)rrrr*rrnrnrorB rMz+ParetoDistribution._characteristic_functionN) rrrrrrrrvrrrrrnrnrnror' s   rcCr)a Create a continuous random variable with the Pareto distribution. Explanation =========== The density of the Pareto distribution is given by .. math:: f(x) := \frac{\alpha\,x_m^\alpha}{x^{\alpha+1}} with :math:`x \in [x_m,\infty]`. Parameters ========== xm : Real number, `x_m > 0`, a scale alpha : Real number, `\alpha > 0`, a shape Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import Pareto, density >>> from sympy import Symbol >>> xm = Symbol("xm", positive=True) >>> beta = Symbol("beta", positive=True) >>> z = Symbol("z") >>> X = Pareto("x", xm, beta) >>> density(X)(z) beta*xm**beta*z**(-beta - 1) References ========== .. [1] https://en.wikipedia.org/wiki/Pareto_distribution .. [2] http://mathworld.wolfram.com/ParetoDistribution.html )rr)rrrrnrnrorYG rrYc@r) PowerFunctionDistributionrrrcCrrjrrrnrnror rzPowerFunctionDistribution.setcCs6t|jdt|jdt||kdt|jddS)Nz-Continuous Boundary parameter should be real.zC 'a' the left Boundary must be smaller than 'b' the right Boundary.z.Continuous Shape parameter should be positive.rrrnrnrorv s  zPowerFunctionDistribution.checkcCs>|j|j|j}}}||||d}|||}||Srr)rrsrrrrrrnrnror s zPowerFunctionDistribution.pdfNrrnrnrnror} s   rcCr)a5 Creates a continuous random variable with a Power Function Distribution. Explanation =========== The density of PowerFunction distribution is given by .. math:: f(x) := \frac{{\alpha}(x - a)^{\alpha - 1}}{(b - a)^{\alpha}} with :math:`x \in [a,b]`. Parameters ========== alpha : Positive number, `0 < \alpha`, the shape paramater a : Real number, :math:`-\infty < a`, the left boundary b : Real number, :math:`a < b < \infty`, the right boundary Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import PowerFunction, density, cdf, E, variance >>> from sympy import Symbol >>> alpha = Symbol("alpha", positive=True) >>> a = Symbol("a", real=True) >>> b = Symbol("b", real=True) >>> z = Symbol("z") >>> X = PowerFunction("X", 2, a, b) >>> density(X)(z) (-2*a + 2*z)/(-a + b)**2 >>> cdf(X)(z) Piecewise((a**2/(a**2 - 2*a*b + b**2) - 2*a*z/(a**2 - 2*a*b + b**2) + z**2/(a**2 - 2*a*b + b**2), a <= z), (0, True)) >>> alpha = 2 >>> a = 0 >>> b = 1 >>> Y = PowerFunction("Y", alpha, a, b) >>> E(Y) 2/3 >>> variance(Y) 1/18 References ========== .. [1] http://www.mathwave.com/help/easyfit/html/analyses/distributions/power_func.html )rr)rrrrrnrnrorZ r4rZc@@eZdZdZeddZeddZddZdd Z d d Z d S) QuadraticUDistributionrcCrrjrrrnrnror rzQuadraticUDistribution.setcCst||kd|dS)Nz&Parameter b must be in range (%s, oo).rrrnrnrorv szQuadraticUDistribution.checkcCsX|j|j}}d||d}||d}t|||dt||k||kftjdfS)N rrT)rrr!r-rr)rrsrrrr rnrnror s  zQuadraticUDistribution.pdfcCs|j|j}}dt||d|dd|d||d|t||dd|||d|||d|dS)NrrxrNr)rrrrOrnrnror s 4&z2QuadraticUDistribution._moment_generating_functioncCsj|j|j}}dttt||tt||dtd|||d|||d|dS)NrrrNrr)rrrrrOrnrnror sFz/QuadraticUDistribution._characteristic_functionN) rrrrrrrrvrrrrnrnrnror s   rcCr)a Create a Continuous Random Variable with a U-quadratic distribution. Explanation =========== The density of the U-quadratic distribution is given by .. math:: f(x) := \alpha (x-\beta)^2 with :math:`x \in [a,b]`. Parameters ========== a : Real number b : Real number, :math:`a < b` Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import QuadraticU, density >>> from sympy import Symbol, pprint >>> a = Symbol("a", real=True) >>> b = Symbol("b", real=True) >>> z = Symbol("z") >>> X = QuadraticU("x", a, b) >>> D = density(X)(z) >>> pprint(D, use_unicode=False) / 2 | / a b \ |12*|- - - - + z| | \ 2 2 / <----------------- for And(b >= z, a <= z) | 3 | (-a + b) | \ 0 otherwise References ========== .. [1] https://en.wikipedia.org/wiki/U-quadratic_distribution )rrrrnrnror[ 8r[c@r) RaisedCosineDistributionrcCst|j|j|j|jSrj)r.r2r@rrnrnror4szRaisedCosineDistribution.setcCr)Nrzs must be positiverrrnrnrorv8ruzRaisedCosineDistribution.checkcCsT|j|j}}tdtt|||d|t|||k|||kftjdfS)NrrT)r2r@r!rrr-rrrrnrnror<s 8zRaisedCosineDistribution.pdfc Cs|j|j}}ttt t||dt|t |fttt||dt|t|ftdt||tt||||td|d|ddfSr)r2r@r!rrrrrrrnrnrorBs *$Fz1RaisedCosineDistribution._characteristic_functioncCsN|j|j}}tdt||t||||td|d|dSr)r2r@rrrrrnrnrorHs@z4RaisedCosineDistribution._moment_generating_functionN rrrrrrrrvrrrrnrnrnror1s   rcCr)a. Create a Continuous Random Variable with a raised cosine distribution. Explanation =========== The density of the raised cosine distribution is given by .. math:: f(x) := \frac{1}{2s}\left(1+\cos\left(\frac{x-\mu}{s}\pi\right)\right) with :math:`x \in [\mu-s,\mu+s]`. Parameters ========== mu : Real number s : Real number, `s > 0` Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import RaisedCosine, density >>> from sympy import Symbol, pprint >>> mu = Symbol("mu", real=True) >>> s = Symbol("s", positive=True) >>> z = Symbol("z") >>> X = RaisedCosine("x", mu, s) >>> D = density(X)(z) >>> pprint(D, use_unicode=False) / /pi*(-mu + z)\ |cos|------------| + 1 | \ s / <--------------------- for And(z >= mu - s, z <= mu + s) | 2*s | \ 0 otherwise References ========== .. [1] https://en.wikipedia.org/wiki/Raised_cosine_distribution )rrrrnrnror\Ls6r\c@rF)RayleighDistributionrrcCr,)Nrz'Scale parameter sigma must be positive.rrrnrnrorvruzRayleighDistribution.checkcCs,|j}||dt|d d|dSrrrrrsrrnrnrors&zRayleighDistribution.pdfcCs$|j}dt|dd|d SrrrrnrnrorszRayleighDistribution._cdfcCsP|j}d||t|d |ddttdt||tdtSr)rrrrr%rrrrrnrnrorrz-RayleighDistribution._characteristic_functioncCsN|j}d||t|d|ddttdt||tddSr)rrrrr#rrnrnrorsHz0RayleighDistribution._moment_generating_functionNr)rnrnrnrors   rcCr)a Create a continuous random variable with a Rayleigh distribution. Explanation =========== The density of the Rayleigh distribution is given by .. math :: f(x) := \frac{x}{\sigma^2} e^{-x^2/2\sigma^2} with :math:`x > 0`. Parameters ========== sigma : Real number, `\sigma > 0` Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import Rayleigh, density, E, variance >>> from sympy import Symbol >>> sigma = Symbol("sigma", positive=True) >>> z = Symbol("z") >>> X = Rayleigh("x", sigma) >>> density(X)(z) z*exp(-z**2/(2*sigma**2))/sigma**2 >>> E(X) sqrt(2)*sqrt(pi)*sigma/2 >>> variance(X) -pi*sigma**2/2 + 2*sigma**2 References ========== .. [1] https://en.wikipedia.org/wiki/Rayleigh_distribution .. [2] http://mathworld.wolfram.com/RayleighDistribution.html )rr)rrrnrnror]s4r]c@r) ReciprocalDistributionrcCrrjrrrnrnrorrzReciprocalDistribution.setcCs,t|dkd|t||kd||fdS)NrzParameter > 0. a = %sz.Parameter b must be in range (%s, +oo]. b = %srrrnrnrorvs zReciprocalDistribution.checkcCs&|j|j}}d|t|t|Sr)rrrrrnrnrorrwzReciprocalDistribution.pdfNrrnrnrnrorrrcCr)aCreates a continuous random variable with a reciprocal distribution. Parameters ========== a : Real number, :math:`0 < a` b : Real number, :math:`a < b` Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import Reciprocal, density, cdf >>> from sympy import symbols >>> a, b, x = symbols('a, b, x', positive=True) >>> R = Reciprocal('R', a, b) >>> density(R)(x) 1/(x*(-log(a) + log(b))) >>> cdf(R)(x) Piecewise((log(a)/(log(a) - log(b)) - log(x)/(log(a) - log(b)), a <= x), (0, True)) Reference ========= .. [1] https://en.wikipedia.org/wiki/Reciprocal_distribution )rrrrnrnror^s"r^c@r)ShiftedGompertzDistributionr_rcCrrarr_rnrnrorvrz!ShiftedGompertzDistribution.checkcCsR|j|j}}|t| |t| t| |d|dt| |Sr)rr`r)rrsrr`rnrnror sDzShiftedGompertzDistribution.pdfNr=rnrnrnror    rcCr)a Create a continuous random variable with a Shifted Gompertz distribution. Explanation =========== The density of the Shifted Gompertz distribution is given by .. math:: f(x) := b e^{-b x} e^{-\eta \exp(-b x)} \left[1 + \eta(1 - e^(-bx)) \right] with :math:`x \in [0, \infty)`. Parameters ========== b : Real number, `b > 0`, a scale eta : Real number, `\eta > 0`, a shape Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import ShiftedGompertz, density >>> from sympy import Symbol >>> b = Symbol("b", positive=True) >>> eta = Symbol("eta", positive=True) >>> x = Symbol("x") >>> X = ShiftedGompertz("x", b, eta) >>> density(X)(x) b*(eta*(1 - exp(-b*x)) + 1)*exp(-b*x)*exp(-eta*exp(-b*x)) References ========== .. [1] https://en.wikipedia.org/wiki/Shifted_Gompertz_distribution )rrrgrnrnror`$s-r`c@s@eZdZdZee eZeddZddZ ddZ dd Z d S) StudentTDistributionnucCr,)Nrz'Degrees of freedom nu must be positive.rrrnrnrorv\ruzStudentTDistribution.checkcCs@|j}dt|ttj|dd|d||d dSr)rrrrrrrsrrnrnror`r zStudentTDistribution.pdfcCsd|j}tj|t|ddttj|ddftddf|d |tt|t|dS)Nrrr)rrrr(r+rrrrrnrnrords (zStudentTDistribution._cdfcCr)NzKThe moment generating function for the Student-T distribution is undefined.rrrnrnrorirz0StudentTDistribution._moment_generating_functionNrfrnrnrnrorWs   rcCr)a0 Create a continuous random variable with a student's t distribution. Explanation =========== The density of the student's t distribution is given by .. math:: f(x) := \frac{\Gamma \left(\frac{\nu+1}{2} \right)} {\sqrt{\nu\pi}\Gamma \left(\frac{\nu}{2} \right)} \left(1+\frac{x^2}{\nu} \right)^{-\frac{\nu+1}{2}} Parameters ========== nu : Real number, `\nu > 0`, the degrees of freedom Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import StudentT, density, cdf >>> from sympy import Symbol, pprint >>> nu = Symbol("nu", positive=True) >>> z = Symbol("z") >>> X = StudentT("x", nu) >>> D = density(X)(z) >>> pprint(D, use_unicode=False) nu 1 - -- - - 2 2 / 2\ | z | |1 + --| \ nu/ ----------------- ____ / nu\ \/ nu *B|1/2, --| \ 2 / >>> cdf(X)(z) 1/2 + z*gamma(nu/2 + 1/2)*hyper((1/2, nu/2 + 1/2), (3/2,), -z**2/nu)/(sqrt(pi)*sqrt(nu)*gamma(nu/2)) References ========== .. [1] https://en.wikipedia.org/wiki/Student_t-distribution .. [2] http://mathworld.wolfram.com/Studentst-Distribution.html )rr)rrrnrnror_ms>r_c@r) TrapezoidalDistributionrrrudcCrrj)r.rrrrnrnrorrzTrapezoidalDistribution.setcCspt||kd||ft||k||kfd|||ft||k||kfd|||ft||kd||fdS)Nz$Lower bound parameter a < %s. a = %sz9Level start parameter b must be in range [%s, %s). b = %sz7Level end parameter c must be in range (%s, %s]. c = %sz$Upper bound parameter d > %s. d = %srrrnrnrorvs  zTrapezoidalDistribution.checkcCs|j|j|j|jf\}}}}td||||||||t||k||kfd||||t||k||kfd||||||||t||k||kftjdfSr)rrrurr!r-rr)rrsrrrurrnrnrors4$4zTrapezoidalDistribution.pdfNrrnrnrnrors   rcCst|t||||fS)a; Create a continuous random variable with a trapezoidal distribution. Explanation =========== The density of the trapezoidal distribution is given by .. math:: f(x) := \begin{cases} 0 & \mathrm{for\ } x < a, \\ \frac{2(x-a)}{(b-a)(d+c-a-b)} & \mathrm{for\ } a \le x < b, \\ \frac{2}{d+c-a-b} & \mathrm{for\ } b \le x < c, \\ \frac{2(d-x)}{(d-c)(d+c-a-b)} & \mathrm{for\ } c \le x < d, \\ 0 & \mathrm{for\ } d < x. \end{cases} Parameters ========== a : Real number, :math:`a < d` b : Real number, :math:`a \le b < c` c : Real number, :math:`b < c \le d` d : Real number Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import Trapezoidal, density >>> from sympy import Symbol, pprint >>> a = Symbol("a") >>> b = Symbol("b") >>> c = Symbol("c") >>> d = Symbol("d") >>> z = Symbol("z") >>> X = Trapezoidal("x", a,b,c,d) >>> pprint(density(X)(z), use_unicode=False) / -2*a + 2*z |------------------------- for And(a <= z, b > z) |(-a + b)*(-a - b + c + d) | | 2 | -------------- for And(b <= z, c > z) < -a - b + c + d | | 2*d - 2*z |------------------------- for And(d >= z, c <= z) |(-c + d)*(-a - b + c + d) | \ 0 otherwise References ========== .. [1] https://en.wikipedia.org/wiki/Trapezoidal_distribution )rr)rrrrurrnrnrorasBrac@r) TriangularDistributionrrrucCrrjrrrnrnrorrzTriangularDistribution.setcCs:t||kd||ft||k||kfd|||fdS)NzParameter b > %s. b = %sz-Parameter c must be in range [%s, %s]. c = %srrrnrnrorvs zTriangularDistribution.checkcCs|j|j|j}}}td||||||t||k||kfd||t||fd||||||t||k||kftjdfSr)rrrur!r-rrr)rrsrrrurnrnrors,,zTriangularDistribution.pdfcCs|j|j|j}}}d||tt||||tt||||tt|||||||||dS)Nrxr)rrrurrrrrrrurnrnror&slz/TriangularDistribution._characteristic_functioncCsv|j|j|j}}}d||t||||t||||t|||||||||dSr)rrrurrrnrnror*s>z2TriangularDistribution._moment_generating_functionNrrnrnrnrors   rcCr)a Create a continuous random variable with a triangular distribution. Explanation =========== The density of the triangular distribution is given by .. math:: f(x) := \begin{cases} 0 & \mathrm{for\ } x < a, \\ \frac{2(x-a)}{(b-a)(c-a)} & \mathrm{for\ } a \le x < c, \\ \frac{2}{b-a} & \mathrm{for\ } x = c, \\ \frac{2(b-x)}{(b-a)(b-c)} & \mathrm{for\ } c < x \le b, \\ 0 & \mathrm{for\ } b < x. \end{cases} Parameters ========== a : Real number, :math:`a \in \left(-\infty, \infty\right)` b : Real number, :math:`a < b` c : Real number, :math:`a \leq c \leq b` Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import Triangular, density >>> from sympy import Symbol, pprint >>> a = Symbol("a") >>> b = Symbol("b") >>> c = Symbol("c") >>> z = Symbol("z") >>> X = Triangular("x", a,b,c) >>> pprint(density(X)(z), use_unicode=False) / -2*a + 2*z |----------------- for And(a <= z, c > z) |(-a + b)*(-a + c) | | 2 | ------ for c = z < -a + b | | 2*b - 2*z |---------------- for And(b >= z, c < z) |(-a + b)*(b - c) | \ 0 otherwise References ========== .. [1] https://en.wikipedia.org/wiki/Triangular_distribution .. [2] http://mathworld.wolfram.com/TriangularDistribution.html )rr)rrrrurnrnrorb0sBrbc@sPeZdZdZeddZeddZddZdd Z d d Z d d Z ddZ dS)UniformDistributionrrcCrrjrrrnrnror{rzUniformDistribution.setcCst||kddS)Nz,Lower limit should be less than Upper limit.rrrnrnrorvruzUniformDistribution.checkcCs:|j|j}}ttj||t||k||kftjdfSrD)rrr!rrr-rrrsrrrnrnrors zUniformDistribution.pdfcCs>|j|j}}ttj||kf||||||kftjdfSrD)rrr!rrrrrnrnrors  zUniformDistribution._cdfcCsT|j|j}}ttt||tt||t|||t|dftjdfSr)rrr!rrrrrrrrrrnrnrors:z,UniformDistribution._characteristic_functioncCsH|j|j}}tt||t|||||t|dftjdfSr)rrr!rrrrrrnrnrors.z/UniformDistribution._moment_generating_functioncKsLd|d<tj|||fi|}|t|j|j|jt|j|j|ji}|S)NTevaluate)r1rsubsrrrr )rrrrresultrnrnrors zUniformDistribution.expectationN) rrrrrrrrvrrrrrrnrnrnrorxs   rcCr)a  Create a continuous random variable with a uniform distribution. Explanation =========== The density of the uniform distribution is given by .. math:: f(x) := \begin{cases} \frac{1}{b - a} & \text{for } x \in [a,b] \\ 0 & \text{otherwise} \end{cases} with :math:`x \in [a,b]`. Parameters ========== a : Real number, :math:`-\infty < a`, the left boundary b : Real number, :math:`a < b < \infty`, the right boundary Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import Uniform, density, cdf, E, variance >>> from sympy import Symbol, simplify >>> a = Symbol("a", negative=True) >>> b = Symbol("b", positive=True) >>> z = Symbol("z") >>> X = Uniform("x", a, b) >>> density(X)(z) Piecewise((1/(-a + b), (b >= z) & (a <= z)), (0, True)) >>> cdf(X)(z) Piecewise((0, a > z), ((-a + z)/(-a + b), b >= z), (1, True)) >>> E(X) a/2 + b/2 >>> simplify(variance(X)) a**2/12 - a*b/6 + b**2/12 References ========== .. [1] https://en.wikipedia.org/wiki/Uniform_distribution_%28continuous%29 .. [2] http://mathworld.wolfram.com/UniformDistribution.html )rr)rrrrnrnrorcs 0` Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import UniformSum, density, cdf >>> from sympy import Symbol, pprint >>> n = Symbol("n", integer=True) >>> z = Symbol("z") >>> X = UniformSum("x", n) >>> D = density(X)(z) >>> pprint(D, use_unicode=False) floor(z) ___ \ ` \ k n - 1 /n\ ) (-1) *(-k + z) *| | / \k/ /__, k = 0 -------------------------------- (n - 1)! >>> cdf(X)(z) Piecewise((0, z < 0), (Sum((-1)**_k*(-_k + z)**n*binomial(n, _k), (_k, 0, floor(z)))/factorial(n), n >= z), (1, True)) Compute cdf with specific 'x' and 'n' values as follows : >>> cdf(UniformSum("x", 5), evaluate=False)(2).doit() 9/40 The argument evaluate=False prevents an attempt at evaluation of the sum for general n, before the argument 2 is passed. References ========== .. [1] https://en.wikipedia.org/wiki/Uniform_sum_distribution .. [2] http://mathworld.wolfram.com/UniformSumDistribution.html )rr)rrrnrnrordsFrdc@s2eZdZdZeddeZeddZddZ dS) VonMisesDistributionr2rrrcCr)NrrIrrrnrnrorvWruzVonMisesDistribution.checkcCs4|j|j}}t|t||dttd|S)Nrr)r2rrrrr)rrsr2rrnrnror[rzVonMisesDistribution.pdfN) rrrrr.rrrrvrrnrnrnrorRs   rcCr)a Create a Continuous Random Variable with a von Mises distribution. Explanation =========== The density of the von Mises distribution is given by .. math:: f(x) := \frac{e^{\kappa\cos(x-\mu)}}{2\pi I_0(\kappa)} with :math:`x \in [0,2\pi]`. Parameters ========== mu : Real number Measure of location. k : Real number Measure of concentration. Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import VonMises, density >>> from sympy import Symbol, pprint >>> mu = Symbol("mu") >>> k = Symbol("k", positive=True) >>> z = Symbol("z") >>> X = VonMises("x", mu, k) >>> D = density(X)(z) >>> pprint(D, use_unicode=False) k*cos(mu - z) e ------------------ 2*pi*besseli(0, k) References ========== .. [1] https://en.wikipedia.org/wiki/Von_Mises_distribution .. [2] http://mathworld.wolfram.com/vonMisesDistribution.html )rr)rr2rrnrnrore_rrec@r)WeibullDistributionrrcCr)NrrzBeta must be positiverrrnrnrorvrzWeibullDistribution.checkcCs8|j|j}}||||dt||| |Sr)rr rrrnrnrorrKzWeibullDistribution.pdfNr=rnrnrnrorrrcCr)a Create a continuous random variable with a Weibull distribution. Explanation =========== The density of the Weibull distribution is given by .. math:: f(x) := \begin{cases} \frac{k}{\lambda}\left(\frac{x}{\lambda}\right)^{k-1} e^{-(x/\lambda)^{k}} & x\geq0\\ 0 & x<0 \end{cases} Parameters ========== lambda : Real number, $\lambda > 0$, a scale k : Real number, $k > 0$, a shape Returns ======= RandomSymbol Examples ======== >>> from sympy.stats import Weibull, density, E, variance >>> from sympy import Symbol, simplify >>> l = Symbol("lambda", positive=True) >>> k = Symbol("k", positive=True) >>> z = Symbol("z") >>> X = Weibull("x", l, k) >>> density(X)(z) k*(z/lambda)**(k - 1)*exp(-(z/lambda)**k)/lambda >>> simplify(E(X)) lambda*gamma(1 + 1/k) >>> simplify(variance(X)) lambda**2*(-gamma(1 + 1/k)**2 + gamma(1 + 2/k)) References ========== .. [1] https://en.wikipedia.org/wiki/Weibull_distribution .. [2] http://mathworld.wolfram.com/WeibullDistribution.html )rrrrnrnrorgrrgc@r) WignerSemicircleDistributionRcCst|j |jSrj)r.rrrnrnrorsz WignerSemicircleDistribution.setcCr,)NrzRadius R must be positive.rrrnrnrorvruz"WignerSemicircleDistribution.checkcCs*|j}dt|dt|d|dSr)rrr)rrsrrnrnrors$z WignerSemicircleDistribution.pdfcC4tdtd|j||j|t|dftjdfSNrrrT)r!rrrrrrrnrnror(z5WignerSemicircleDistribution._characteristic_functioncCrr)r!rrrrrrrnrnrorrz8WignerSemicircleDistribution._moment_generating_functionNrrnrnrnrors   rcCr)af Create a continuous random variable with a Wigner semicircle distribution. Explanation =========== The density of the Wigner semicircle distribution is given by .. math:: f(x) := \frac2{\pi R^2}\,\sqrt{R^2-x^2} with :math:`x \in [-R,R]`. Parameters ========== R : Real number, `R > 0`, the radius Returns ======= A RandomSymbol. Examples ======== >>> from sympy.stats import WignerSemicircle, density, E >>> from sympy import Symbol >>> R = Symbol("R", positive=True) >>> z = Symbol("z") >>> X = WignerSemicircle("x", R) >>> density(X)(z) 2*sqrt(R**2 - z**2)/(pi*R**2) >>> E(X) 0 References ========== .. [1] https://en.wikipedia.org/wiki/Wigner_semicircle_distribution .. [2] http://mathworld.wolfram.com/WignersSemicircleLaw.html )rr)rrrnrnrorhs1rhN)rrrE)F)__doc__&sympy.functions.elementary.exponentialr(sympy.functions.elementary.trigonometricrrrrsympy.functions.special.besselrrr &sympy.functions.special.beta_functionsr rsympy.concrete.summationsr sympy.core.basicr sympy.core.functionr sympy.core.numbersrrrsympy.core.relationalrrsympy.core.singletonrsympy.core.symbolrsympy.core.sympifyr(sympy.functions.combinatorial.factorialsrr$sympy.functions.elementary.complexesrrr%sympy.functions.elementary.hyperbolicr#sympy.functions.elementary.integersr(sympy.functions.elementary.miscellaneousrrr $sympy.functions.elementary.piecewiser!r"'sympy.functions.special.error_functionsr#r$r%r&r''sympy.functions.special.gamma_functionsr(r)r*sympy.functions.special.hyperr+sympy.integrals.integralsr,sympy.logic.boolalgr-sympy.sets.setsr.sympy.matricesr/sympy.stats.crvr0r1sympy.stats.rvr2r3Infinityr__all__registerrtrrr4rr5rr6rr7rr8rr9rr:rr;rr<rr=r r>r r?r@rrAr*rBr0rCr5rDr<rEr>rFrrGrLrHrPrJr^rIrirKrlrLrsrMr{rNrrOrrPrrQrrRrrSrrTrrUrrVrrWrrXrfrrYrrZrr[rr\rr]rr^rr`rr_rrarrbrrcrrdrrerrgrrhrnrnrnros <                 8  : 8B=K53148>?I)MOE?B ;OA !@4;E437EAG6:7Lb#D 6D><9)3DHH,B L =>