ó ¡¼™\c@ s'dZddlmZmZddlmZmZmZmZm Z m Z m Z m Z m Z mZmZmZmZmZmZmZmZmZmZmZmZmZmZmZddlmZddlm Z ddl!m"Z"ddl#m$Z$ddl%m&Z&m'Z'm(Z(m)Z)m*Z*m+Z+m,Z,m-Z-dd l.Z.d e&fd „ƒYZ/d e/e'fd „ƒYZ0de)e/fd„ƒYZ1de/e(fd„ƒYZ2defd„ƒYZ3de3e-fd„ƒYZ4de4fd„ƒYZ5de*fd„ƒYZ6de6e+fd„ƒYZ7d„Z8d„Z9d S(sl Continuous Random Variables Module See Also ======== sympy.stats.crv_types sympy.stats.rv sympy.stats.frv iÿÿÿÿ(tprint_functiontdivision(tIntervalt IntersectiontsymbolstsympifytDummytIntegraltAndtOrt Piecewisetcacheitt integratetootLambdatBasictStexptIt FiniteSettNetEqtUniontpolytseriest factorial(t DiracDelta(tPolynomialError(tsolveset(treduce_rational_inequalities(t RandomDomaint SingleDomaintConditionalDomaint ProductDomaintPSpacet SinglePSpacetrandom_symbolstNamedArgsMixinNtContinuousDomaincB seZdZeZd„ZRS(sX A domain with continuous support Represented using symbols and Intervals. cC stdƒ‚dS(Ns#Not Implemented for generic Domains(tNotImplementedError(tself((s.lib/python2.7/site-packages/sympy/stats/crv.pyt as_boolean!s(t__name__t __module__t__doc__tTruet is_ContinuousR)(((s.lib/python2.7/site-packages/sympy/stats/crv.pyR&stSingleContinuousDomaincB s#eZdZdd„Zd„ZRS(sj A univariate domain with continuous support Represented using a single symbol and interval. cK sh|dkr|j}n|s"|St|ƒt|jƒkrLtdƒ‚nt||j|jf|S(NsValues should be equal(tNoneRt frozensett ValueErrorRtsymboltset(R(texprt variablestkwargs((s.lib/python2.7/site-packages/sympy/stats/crv.pytcompute_expectation+s  cC s|jj|jƒS(N(R4t as_relationalR3(R(((s.lib/python2.7/site-packages/sympy/stats/crv.pyR)5sN(R*R+R,R0R8R)(((s.lib/python2.7/site-packages/sympy/stats/crv.pyR/%s tProductContinuousDomaincB s#eZdZdd„Zd„ZRS(sE A collection of independent domains with continuous support cK sg|dkr|j}nxH|jD]=}t|ƒt|jƒ@}|r"|j|||}q"q"W|S(N(R0RtdomainsR1R8(R(R5R6R7tdomaint domain_vars((s.lib/python2.7/site-packages/sympy/stats/crv.pyR8>s  cC s&tg|jD]}|jƒ^q ŒS(N(RR;R)(R(R<((s.lib/python2.7/site-packages/sympy/stats/crv.pyR)GsN(R*R+R,R0R8R)(((s.lib/python2.7/site-packages/sympy/stats/crv.pyR:9s tConditionalContinuousDomaincB s2eZdZdd„Zd„Zed„ƒZRS(sl A domain with continuous support that has been further restricted by a condition such as x > 3 cK sÔ|dkr|j}n|s"|S|jj||ƒ}|jt|jƒ}}|jg}xe|rÃ|jƒ}|j r½t |t ƒrœ|j |j ƒqÀt |tƒrÀtdƒ‚qÀq_|jr°|jrì|t|j|jƒ9}qÀ|jt|jƒ@} t| ƒdkr#tdƒ‚n| jƒ} xŽt|ƒD]m\} } | d| kr<t|| ƒ} t| d| dƒ}| j|ƒ}| |j|jf|| Ks / tContinuousDistributioncB seZd„ZRS(cG s |j|ŒS(N(tpdf(R(RH((s.lib/python2.7/site-packages/sympy/stats/crv.pyt__call__Žs(R*R+Rb(((s.lib/python2.7/site-packages/sympy/stats/crv.pyR`stSingleContinuousDistributioncB s½eZdZee eƒZd„Zed„ƒZd„Z e d„ƒZ e d„ƒZ d„Z d„Ze d„ƒZd „Zd „Ze d „ƒZd „Zd „Zed„ZRS(sv Continuous distribution of a single variable Serves as superclass for Normal/Exponential/UniformDistribution etc.... Represented by parameters for each of the specific classes. E.g NormalDistribution is represented by a mean and standard deviation. Provides methods for pdf, cdf, and sampling See Also ======== sympy.stats.crv_types.* cG s%ttt|ƒƒ}tj||ŒS(N(RAtmapRRt__new__(tclsRH((s.lib/python2.7/site-packages/sympy/stats/crv.pyRe¤scG sdS(N((RH((s.lib/python2.7/site-packages/sympy/stats/crv.pytcheck¨scC s"|jƒ}|tjddƒƒS(s, A random realization from the distribution ii(t_inverse_cdf_expressiontrandomtuniform(R(ticdf((s.lib/python2.7/site-packages/sympy/stats/crv.pytsample¬s cC sÔtddtdtdtƒ\}}y]t|j|ƒ||tjƒ}t|tƒr€tj|j kr€t |j dƒ}nWnt k ršd}nX| s´t |ƒdkrÃt dƒ‚nt||dƒS( s4 Inverse of the CDF Used by sample sx, ztrealtpositiveRfisCould not invert CDFiN(RR-RRtcdfRtRealsRFRRHRAR'R0RNR(R(txtzt inverse_cdf((s.lib/python2.7/site-packages/sympy/stats/crv.pyRh±s$"!  cK sˆtddtdtdtƒ\}}|jj}|j|ƒ}t||||f|}t|||kfdtfƒ}t||ƒS(s@ Compute the CDF from the PDF Returns a Lambda sx, zRmtfiniteRfi( RR-RR4tstartRaR R R(R(R7RqRrt left_boundRaRo((s.lib/python2.7/site-packages/sympy/stats/crv.pyt compute_cdfÄs $ !cC sdS(N(R0(R(Rq((s.lib/python2.7/site-packages/sympy/stats/crv.pyt_cdfÔscK sGt|ƒdkr4|j|ƒ}|dk r4|Sn|j||ƒS(s Cumulative density function iN(RNRxR0Rw(R(RqR7Ro((s.lib/python2.7/site-packages/sympy/stats/crv.pyRo×s  cK sktddtdtdtƒ\}}|j|ƒ}ttt||ƒ||t tfƒ}t||ƒS(sT Compute the characteristic function from the PDF Returns a Lambda sx, tRmRtRf( RR-RRaR RRR R(R(R7RqttRatcf((s.lib/python2.7/site-packages/sympy/stats/crv.pytcompute_characteristic_functionßs$+cC sdS(N(R0(R(Ry((s.lib/python2.7/site-packages/sympy/stats/crv.pyt_characteristic_functionêscK sGt|ƒdkr4|j|ƒ}|dk r4|Sn|j||ƒS(s Characteristic function iN(RNR|R0R{(R(RyR7Rz((s.lib/python2.7/site-packages/sympy/stats/crv.pytcharacteristic_functionís  cK satddtdtƒ\}}|j|ƒ}tt||ƒ||t tfƒ}t||ƒS(sW Compute the moment generating function from the PDF Returns a Lambda sx, tRmRf(RR-RRaR RR R(R(R7RqRyRatmgf((s.lib/python2.7/site-packages/sympy/stats/crv.pyt"compute_moment_generating_functionõs'cC sdS(N(R0(R(Ry((s.lib/python2.7/site-packages/sympy/stats/crv.pyt_moment_generating_functionscK s`t|ƒdkrMy#|j|ƒ}|dk r4|SWqMtk rIdSXn|j||ƒS(s Moment generating function iN(RNR€R0R'R(R(RyR7R~((s.lib/python2.7/site-packages/sympy/stats/crv.pytmoment_generating_functions  c K s[|r1yñt||ƒ}tddtƒ}|j|ƒ}|dkrkt||j|ƒ||jf|S|jƒ}tt ||d|dƒj ƒ|ƒ} d} xJt |dƒD]8} | |j || ƒ| j || ƒt | ƒ7} q¶W| SWqWtk r-t||j|ƒ||jf|SXn&t||j|ƒ||jf|SdS(s- Expectation of expression over distribution RyRmiiN(RRR-R€R0R RaR4tdegreeRtremoveOtrangetcoeff_monomialRRR( R(R5tvartevaluateR7tpRyR~tdegttaylortresulttk((s.lib/python2.7/site-packages/sympy/stats/crv.pyt expectations  & (6 *(R*R+R,RR R4Ret staticmethodRgRlR RhRwRxRoR{R|R}RR€RR-R(((s.lib/python2.7/site-packages/sympy/stats/crv.pyRc’s         tContinuousDistributionHandmadecB s3eZdZed„ƒZee eƒd„ZRS(RacC s |jdS(Ni(RH(R(((s.lib/python2.7/site-packages/sympy/stats/crv.pyR4&scC stj|||ƒS(N(RRe(RfRaR4((s.lib/python2.7/site-packages/sympy/stats/crv.pyRe*s(Ra(R*R+t _argnamesR_R4RR Re(((s.lib/python2.7/site-packages/sympy/stats/crv.pyR#stContinuousPSpacecB sŒeZdZeZeZed„ƒZd e d„Z d„Z e d„ƒZ e d„ƒZe d„ƒZd„Zd„Zed „ZRS( s® Continuous Probability Space Represents the likelihood of an event space defined over a continuum. Represented with a ContinuousDomain and a PDF (Lambda-Like) cC s|j|jjŒS(N(tdensityR<R(R(((s.lib/python2.7/site-packages/sympy/stats/crv.pyRa9scK sv|dkr|j}n t|ƒ}|jtd„|Dƒƒƒ}td„|Dƒƒ}|jj|j|||S(Ncs s|]}||jfVqdS(N(R3(t.0trv((s.lib/python2.7/site-packages/sympy/stats/crv.pys Cscs s|]}|jVqdS(N(R3(R“R”((s.lib/python2.7/site-packages/sympy/stats/crv.pys Es(R0tvaluesR1txreplacetdictR<R8Ra(R(R5trvsR‡R7tdomain_symbols((s.lib/python2.7/site-packages/sympy/stats/crv.pyR8=s   cK s­||jkrrtt|jƒt|gƒƒ}td„|Dƒƒ}|jj|j||}t|j|ƒSt ddt dt ƒ}t||jt ||ƒ|ƒS(Ncs s|]}|jVqdS(N(R3(R“trs((s.lib/python2.7/site-packages/sympy/stats/crv.pys OsRrRmRt( R•R^R4R1R<R8RaRR3RR-R(R(R5R7t randomsymbolsRRaRr((s.lib/python2.7/site-packages/sympy/stats/crv.pytcompute_densityJs"cK s²|jjjstdƒ‚n|j||}tddtdtdtƒ\}}|jjj}t ||ƒ|||f|}t |||kfdtfƒ}t ||ƒS(Ns0CDF not well defined on multivariate expressionssx, zRmRtRfi( R<R4t is_IntervalR2RœRR-RRuR R R(R(R5R7tdRqRrRvRo((s.lib/python2.7/site-packages/sympy/stats/crv.pyRwVs $!!cK s|jjjstdƒ‚n|j||}tddtdtƒ\}}tt t ||ƒ||ƒ|t t f|}t ||ƒS(NsCCharacteristic function of multivariate expressions not implementedsx, tRmRf( R<R4RR'RœRR-RR RRR R(R(R5R7RžRqRyRz((s.lib/python2.7/site-packages/sympy/stats/crv.pyR{fs 4cK s‹|jjjstdƒ‚n|j||}tddtdtƒ\}}tt ||ƒ||ƒ|t t f|}t ||ƒS(NsFMoment generating function of multivariate expressions not implementedsx, tRmRf( R<R4RR'RœRR-RR RR R(R(R5R7RžRqRyR~((s.lib/python2.7/site-packages/sympy/stats/crv.pyRps 0c  sÝtddtdtƒ‰t}t|tƒrSt|jd|jdƒ}t}nyæ|j|ƒ}g|jD]}|j |j kro|^qod}|j |ˆ‰|j t j ksÍt|j tƒrá|sÚt jSt jSt|j tƒrt‡‡‡fd†|j jDƒƒStˆˆƒˆ|j fˆSWntk rØddlm}|j|j}||ˆ}t|tƒs–t|ƒ}ntˆ|ƒ} | j|j| jdƒƒ} |sÍ| St j| SXdS( NRrRmRtiic3 s<|]2}t|tƒrtˆˆƒˆ|fˆVqdS(N(RFRR(R“tsubset(R7RaRr(s.lib/python2.7/site-packages/sympy/stats/crv.pys ‹siÿÿÿÿ(R’(RR-tFalseRFRRRHtwhereR•R3RœR4RtEmptySetRtZerotOneRtsumRR'tsympy.stats.rvR’RKRLR`RtSingleContinuousPSpacet probabilityt __class__tvalue( R(RCR7tcond_invR<R”R’R5tdenstspaceR‹((R7RaRrs.lib/python2.7/site-packages/sympy/stats/crv.pyR¨zs2 2$# cC s‰tt|ƒƒ}t|ƒdko3|j|jƒsEtdƒ‚nt|ƒd}t||ƒ}|j|j j ƒ}t |j |ƒS(Nis2Multiple continuous random variables not supportedi( R1R$RNtissubsetR•R'R^RPRQR<R4R/R3(R(RCR˜R”tinterval((s.lib/python2.7/site-packages/sympy/stats/crv.pyR¡s$ c K sš|jtd„|jDƒƒƒ}t|j|ƒ}|rd„|jDƒ}|j|j|}|j|j|ƒ}t|j|ƒ}nt ||ƒS(Ncs s|]}||jfVqdS(N(R3(R“R”((s.lib/python2.7/site-packages/sympy/stats/crv.pys ¨scS s%i|]}tt|ƒƒ|“qS((Rtstr(R“R”((s.lib/python2.7/site-packages/sympy/stats/crv.pys °s ( R–R—R•R>R<RR8RaRR‘( R(RCt normalizeR7R<t replacementtnormRaR’((s.lib/python2.7/site-packages/sympy/stats/crv.pytconditional_space§s"N(R*R+R,R-R.tis_realR_RaR0R R8RœR RwR{RR¨R¡R´(((s.lib/python2.7/site-packages/sympy/stats/crv.pyR‘.s   # R§cB sheZdZed„ƒZed„ƒZd„Zd ed„Z d„Z d„Z d„Z d„Z RS( s  A continuous probability space over a single univariate variable These consist of a Symbol and a SingleContinuousDistribution This class is normally accessed through the various random variable functions, Normal, Exponential, Uniform, etc.... cC s |jjS(N(t distributionR4(R(((s.lib/python2.7/site-packages/sympy/stats/crv.pyR4ÂscC stt|jƒ|jƒS(N(R/RR3R4(R(((s.lib/python2.7/site-packages/sympy/stats/crv.pyR<ÆscC si|jjƒ|j6S(so Internal sample method Returns dictionary mapping RandomSymbol to realization value. (R¶RlRª(R(((s.lib/python2.7/site-packages/sympy/stats/crv.pyRlÊscK s¥|p|jf}|j|kr%|S|jtd„|Dƒƒƒ}|jj}y |jj||d||SWn.tk r t||j||j f|SXdS(Ncs s|]}||jfVqdS(N(R3(R“R”((s.lib/python2.7/site-packages/sympy/stats/crv.pys ×sR‡( RªR–R—R3R¶Rt ExceptionRRaR4(R(R5R˜R‡R7Rq((s.lib/python2.7/site-packages/sympy/stats/crv.pyR8Òs   cK s`||jkrItddtdtdtƒ}t||jj||ƒStj|||SdS(NRrRmRtRf( RªRR-RRR¶RoR‘Rw(R(R5R7Rr((s.lib/python2.7/site-packages/sympy/stats/crv.pyRwßscK sZ||jkrCtddtdtƒ}t||jj||ƒStj|||SdS(NRyRmRf( RªRR-RRR¶R}R‘R{(R(R5R7Ry((s.lib/python2.7/site-packages/sympy/stats/crv.pyR{æscK sZ||jkrCtddtdtƒ}t||jj||ƒStj|||SdS(NRyRmRf( RªRR-RRR¶RR‘R(R(R5R7Ry((s.lib/python2.7/site-packages/sympy/stats/crv.pyRísc sÕ||jkr|jStdƒ‰t|ˆ|jtjƒ}t|tƒrutj|jkrut |jdƒ}n|s—t d||jfƒ‚n|j |jƒ‰t ‡‡fd†|Dƒƒ}t ˆ|ƒS(NtyisCan not solve %s for %sc3 s.|]$}ˆ|ƒt|jˆƒƒVqdS(N(tabstdiff(R“tg(tfxR¸(s.lib/python2.7/site-packages/sympy/stats/crv.pys s(RªR’RRRRpRFRRHRAR2RœR¥R(R(R5R7tgstfy((R¼R¸s.lib/python2.7/site-packages/sympy/stats/crv.pyRœôs !N(R*R+R,R_R4R<RlR0R R8RwR{RRœ(((s.lib/python2.7/site-packages/sympy/stats/crv.pyR§¸s    cK s@yt|||SWn%tk r;td|dƒ‚nXdS(Ns!Reduction of condition failed %s i(RRR2(RWR†R7((s.lib/python2.7/site-packages/sympy/stats/crv.pyt_reduce_inequalitiess cC sØ|jr"t|gg|dtƒSt|tƒrftg|jD]!}t|gg|dtƒ^q>ŒSt|tƒrÔg|jD]!}t|gg|dtƒ^q}|d}x|D]}|j|ƒ}q·W|SdS(Nt relationali( RIR¿R RFR RRHRRQ(RCR†targt intervalsRRY((s.lib/python2.7/site-packages/sympy/stats/crv.pyRP s /.  (:R,t __future__RRtsympyRRRRRRRR R R R R RRRRRRRRRRRRt'sympy.functions.special.delta_functionsRtsympy.polys.polyerrorsRtsympy.solvers.solvesetRtsympy.solvers.inequalitiesRR¦RRR R!R"R#R$R%RiR&R/R:R>R`RcRR‘R§R¿RP(((s.lib/python2.7/site-packages/sympy/stats/crv.pyt s&š:  B‘ ŠM