ó ¡¼™\c@ sÕdZddlmZmZddlmZddlmZmZm Z m Z m Z m Z m Z mZmZmZmZmZmZmZddlmZddlmZmZmZmZmZmZmZmZm Z m!Z!ddl"m#Z#ddl$Z$d e%fd „ƒYZ&d efd „ƒYZ'd e'fd„ƒYZ(dee'fd„ƒYZ)dee)fd„ƒYZ*dee!fd„ƒYZ+defd„ƒYZ,dee,fd„ƒYZ-dee,fd„ƒYZ.dS(sq Finite Discrete Random Variables Module See Also ======== sympy.stats.frv_types sympy.stats.rv sympy.stats.crv iÿÿÿÿ(tprint_functiontdivision(tproduct(tBasictSymboltcacheittsympifytMultAndtOrtTuplet PiecewisetEqtLambdatexptItDummy(t FiniteSet( t RandomDomaint ProductDomaintConditionalDomaintPSpacetIndependentProductPSpacet SinglePSpacetrandom_symbolstsumsetstrv_substNamedArgsMixin(tDictNt FiniteDensitycB s&eZdZd„Zed„ƒZRS(s' A domain with Finite Density. cC s(t|ƒ}||kr ||SdSdS(s– Make instance of a class callable. If item belongs to current instance of a class, return it. Otherwise, return 0. iN(R(tselftitem((s.lib/python2.7/site-packages/sympy/stats/frv.pyt__call__s  cC s t|ƒS(s, Return item as dictionary. (tdict(R((s.lib/python2.7/site-packages/sympy/stats/frv.pyR!)s(t__name__t __module__t__doc__R tpropertyR!(((s.lib/python2.7/site-packages/sympy/stats/frv.pyRs t FiniteDomaincB s\eZdZeZed„ƒZed„ƒZed„ƒZd„Z d„Z d„Z RS(sS A domain with discrete finite support Represented using a FiniteSet. cC std„|jDƒƒS(Ncs s|]\}}|VqdS(N((t.0tsymtval((s.lib/python2.7/site-packages/sympy/stats/frv.pys :s(Rtelements(R((s.lib/python2.7/site-packages/sympy/stats/frv.pytsymbols8scC s |jdS(Ni(targs(R((s.lib/python2.7/site-packages/sympy/stats/frv.pyR*<scC s,tg|jD]}tt|ƒƒ^q ŒS(N(RR*RR!(Rtel((s.lib/python2.7/site-packages/sympy/stats/frv.pyR!@scC s ||jkS(N(R*(Rtother((s.lib/python2.7/site-packages/sympy/stats/frv.pyt __contains__DscC s |jjƒS(N(R*t__iter__(R((s.lib/python2.7/site-packages/sympy/stats/frv.pyR0Gsc C sEtg|D]4}tg|D]\}}t||ƒ^qŒ^q ŒS(N(R RR (RRR(R)((s.lib/python2.7/site-packages/sympy/stats/frv.pyt as_booleanJs( R"R#R$tTruet is_FiniteR%R+R*R!R/R0R1(((s.lib/python2.7/site-packages/sympy/stats/frv.pyR&0s  tSingleFiniteDomaincB seeZdZd„Zed„ƒZed„ƒZed„ƒZed„ƒZd„Z d„Z RS(si A FiniteDomain over a single symbol/set Example: The possibilities of a *single* die roll. cC s1t|tƒst|Œ}ntj|||ƒS(N(t isinstanceRRt__new__(tclstsymboltset((s.lib/python2.7/site-packages/sympy/stats/frv.pyR6UscC s|jdSt|jƒdS(Ni(R,ttupleR+(R((s.lib/python2.7/site-packages/sympy/stats/frv.pyR8Zs cC s t|jƒS(N(RR8(R((s.lib/python2.7/site-packages/sympy/stats/frv.pyR+_scC s |jdS(Ni(R,(R((s.lib/python2.7/site-packages/sympy/stats/frv.pyR9cscC s2tg|jD]}t|j|ffƒ^q ŒS(N(RR9t frozensetR8(Rtelem((s.lib/python2.7/site-packages/sympy/stats/frv.pyR*gsc s‡fd†ˆjDƒS(Nc3 s'|]}tˆj|ffƒVqdS(N(R;R8(R'R<(R(s.lib/python2.7/site-packages/sympy/stats/frv.pys ls(R9(R((Rs.lib/python2.7/site-packages/sympy/stats/frv.pyR0kscC s2t|ƒd\}}||jko1||jkS(Ni(R:R8R9(RR.R(R)((s.lib/python2.7/site-packages/sympy/stats/frv.pyR/ns( R"R#R$R6R%R8R+R9R*R0R/(((s.lib/python2.7/site-packages/sympy/stats/frv.pyR4Ns  tProductFiniteDomaincB s&eZdZd„Zed„ƒZRS(sŠ A Finite domain consisting of several other FiniteDomains Example: The possibilities of the rolls of three independent dice cC st|jŒ}d„|DƒS(Ncs s|]}t|ƒVqdS(N(R(R'titems((s.lib/python2.7/site-packages/sympy/stats/frv.pys |s(Rtdomains(Rtproditer((s.lib/python2.7/site-packages/sympy/stats/frv.pyR0zscC s t|ŒS(N(R(R((s.lib/python2.7/site-packages/sympy/stats/frv.pyR*~s(R"R#R$R0R%R*(((s.lib/python2.7/site-packages/sympy/stats/frv.pyR=ss tConditionalFiniteDomaincB sJeZdZd„Zd„Zd„Zd„Zed„ƒZd„Z RS(s A FiniteDomain that has been restricted by a condition Example: The possibilities of a die roll under the condition that the roll is even. cC sq|tkr|St|ƒ}|jj|jƒs^td|t|j|jƒfdƒ‚ntj|||ƒS(sH Create a new instance of ConditionalFiniteDomain class s-Condition "%s" contains foreign symbols %s. s.Will be unable to iterate using this condition(R2Rt free_symbolstissubsett ValueErrorR:RR6(R7tdomaint conditiontcond((s.lib/python2.7/site-packages/sympy/stats/frv.pyR6‹s   cC sa|jjt|ƒƒ}|ttgkr.|S|jrG|j|jkStdt |ƒƒ‚dS(sï Test the value. If value is boolean, return it. If value is equality relational (two objects are equal), return it with left-hand side being equal to right-hand side. Otherwise, raise ValueError exception. sUndeciable if %sN( RFtxreplaceR!R2tFalset is_EqualitytlhstrhsRDtstr(RR<R)((s.lib/python2.7/site-packages/sympy/stats/frv.pyt_testžs  cC s||jko|j|ƒS(N(t fulldomainRN(RR.((s.lib/python2.7/site-packages/sympy/stats/frv.pyR/«sc s‡fd†ˆjDƒS(Nc3 s$|]}ˆj|ƒr|VqdS(N(RN(R'R<(R(s.lib/python2.7/site-packages/sympy/stats/frv.pys ¯s(RO(R((Rs.lib/python2.7/site-packages/sympy/stats/frv.pyR0®scC sft|jtƒrVtg|jjD]-}t|jj|ffƒ|kr"|^q"ŒStdƒ‚dS(Ns7Not implemented on multi-dimensional conditional domain(R5ROR4RR9R;R8tNotImplementedError(RR<((s.lib/python2.7/site-packages/sympy/stats/frv.pyR9±s .cC s tj|ƒS(N(R&R1(R((s.lib/python2.7/site-packages/sympy/stats/frv.pyR1ºs( R"R#R$R6RNR/R0R%R9R1(((s.lib/python2.7/site-packages/sympy/stats/frv.pyRAƒs    tSingleFiniteDistributioncB s­eZd„Zeed„ƒƒZed„ƒZed„ƒZed„ƒZed„ƒZ ed„ƒZ ed„ƒZ ed„ƒZ ed „ƒZ eZd „ZRS( cG s%ttt|ƒƒ}tj||ŒS(N(tlisttmapRRR6(R7R,((s.lib/python2.7/site-packages/sympy/stats/frv.pyR6¾sc st‡fd†ˆjDƒƒS(Nc3 s$|]}|ˆj|ƒfVqdS(N(tpdf(R'tk(R(s.lib/python2.7/site-packages/sympy/stats/frv.pys Ås(R!R9(R((Rs.lib/python2.7/site-packages/sympy/stats/frv.pyR!Âsc C s]tdƒ}t|tg|jjƒD]!\}}|t||ƒf^q%dtfgŒƒS(Ntxi(RR R R!R>R R2(RRVRUtv((s.lib/python2.7/site-packages/sympy/stats/frv.pyRTÇs  c s>tddtƒ‰tˆt‡fd†|jjƒDƒƒƒS(Ntttrealc3 s-|]#\}}tt|ˆƒ|VqdS(N(RR(R'RURW(RX(s.lib/python2.7/site-packages/sympy/stats/frv.pys Ðs(RR2R tsumR!R>(R((RXs.lib/python2.7/site-packages/sympy/stats/frv.pytcharacteristic_functionÍsc s>tddtƒ‰tˆt‡fd†|jjƒDƒƒƒS(NRXRYc3 s)|]\}}t|ˆƒ|VqdS(N(R(R'RURW(RX(s.lib/python2.7/site-packages/sympy/stats/frv.pys Õs(RR2R RZR!R>(R((RXs.lib/python2.7/site-packages/sympy/stats/frv.pytmoment_generating_functionÒscC st|jjƒƒS(N(RRR!tkeys(R((s.lib/python2.7/site-packages/sympy/stats/frv.pyR9×scC s |jjS(N(R!tvalues(R((s.lib/python2.7/site-packages/sympy/stats/frv.pytÛtcC s |jjS(N(R!R>(R((s.lib/python2.7/site-packages/sympy/stats/frv.pyR_ÜR`cC s |jjS(N(R!R0(R((s.lib/python2.7/site-packages/sympy/stats/frv.pyR_ÝR`cC s |jjS(N(R!t __getitem__(R((s.lib/python2.7/site-packages/sympy/stats/frv.pyR_ÞR`cC s ||jkS(N(R9(RR.((s.lib/python2.7/site-packages/sympy/stats/frv.pyR/âs(R"R#R6R%RR!RTR[R\R9R^R>R0RaR R/(((s.lib/python2.7/site-packages/sympy/stats/frv.pyRQ½s t FinitePSpacecB sžeZdZeZd„Zd„Zd„Zd„Ze d„ƒZ e e d„ƒZ e d„ƒZ e d„ƒZd d „Zd „Zd „Zd „ZRS(sd A Finite Probability Space Represents the probabilities of a finite number of events. cC sJtd„|jƒDƒƒ}t|ƒ}tj|||ƒ}||_|S(Ncs s-|]#\}}t|ƒt|ƒfVqdS(N(R(R'tkeyR)((s.lib/python2.7/site-packages/sympy/stats/frv.pys ôs(R!R>RRR6t_density(R7REtdensitytpublic_densitytobj((s.lib/python2.7/site-packages/sympy/stats/frv.pyR6ós    cC st|ƒ}|jj|dƒS(Ni(RRdtget(RR<((s.lib/python2.7/site-packages/sympy/stats/frv.pytprob_ofüs c s8t‡fd†t|ƒDƒƒs(t‚tˆj|ƒS(Nc3 s!|]}|jˆjkVqdS(N(R8R+(R'tr(R(s.lib/python2.7/site-packages/sympy/stats/frv.pys s(tallRtAssertionErrorRARE(RRF((Rs.lib/python2.7/site-packages/sympy/stats/frv.pytwheres(cC s|jtd„|jDƒƒƒ}tƒ}xO|jD]D}|jt|ƒƒ}|j|ƒ}|j|dƒ|||si(RHR!R^RRERiRh(RtexprtdR<R)tprob((s.lib/python2.7/site-packages/sympy/stats/frv.pytcompute_densitys" cC sc|j|ƒ}d}g}x;t|ƒD]-}||}||7}|j||fƒq(Wt|ƒS(Ni(RrtsortedtappendR!(RRoRptcum_probtcdfRcRq((s.lib/python2.7/site-packages/sympy/stats/frv.pyt compute_cdf s  cC sn|j|ƒ}t|jƒƒ}t|dd„ƒ}|rjg|D]\}}|t|ƒf^qC}n|S(NRccS s|dS(Ni((t val_cumprob((s.lib/python2.7/site-packages/sympy/stats/frv.pyR_R`(RwRRR>Rstfloat(RRot python_floatRvR>t sorted_itemsRWRu((s.lib/python2.7/site-packages/sympy/stats/frv.pyt sorted_cdfs+c sJ|j|ƒ}tddtƒ‰tˆt‡fd†|jƒDƒƒƒS(NRXRYc3 s-|]#\}}tt|ˆƒ|VqdS(N(RR(R'RURW(RX(s.lib/python2.7/site-packages/sympy/stats/frv.pys (s(RrRR2R RZR>(RRoRp((RXs.lib/python2.7/site-packages/sympy/stats/frv.pytcompute_characteristic_function#sc sJ|j|ƒ}tddtƒ‰tˆt‡fd†|jƒDƒƒƒS(NRXRYc3 s)|]\}}t|ˆƒ|VqdS(N(R(R'RURW(RX(s.lib/python2.7/site-packages/sympy/stats/frv.pys /s(RrRR2R RZR>(RRoRp((RXs.lib/python2.7/site-packages/sympy/stats/frv.pyt"compute_moment_generating_function*scK sj|p |j}|jtd„|Dƒƒƒ}tg|jD](}|jt|ƒƒ|j|ƒ^q;ƒS(Ncs s|]}||jfVqdS(N(R8(R'Rn((s.lib/python2.7/site-packages/sympy/stats/frv.pys 3s(R^RHR!RZRERi(RRotrvstkwargsR<((s.lib/python2.7/site-packages/sympy/stats/frv.pytcompute_expectation1sc sWtd„t|ƒDƒƒ}|jˆjƒs4t‚t‡fd†ˆj|ƒDƒƒS(Ncs s|]}|jVqdS(N(R8(R'Rn((s.lib/python2.7/site-packages/sympy/stats/frv.pys 8sc3 s|]}ˆj|ƒVqdS(N(Ri(R'R<(R(s.lib/python2.7/site-packages/sympy/stats/frv.pys :s(R;RRCR+RlRZRm(RRFt cond_symbols((Rs.lib/python2.7/site-packages/sympy/stats/frv.pyt probability7sc sS|j|ƒ‰|j|ƒ‰t‡‡fd†|jjƒDƒƒ}tˆ|ƒS(Nc3 s4|]*\}}ˆj|ƒr||ˆfVqdS(N(RN(R'RcR)(RERq(s.lib/python2.7/site-packages/sympy/stats/frv.pys ?s(RmRƒR!RdR>Rb(RRFRe((RERqs.lib/python2.7/site-packages/sympy/stats/frv.pytconditional_space<s cC sˆt|jŒ}|j|dtƒ}tjddƒ}x9|D]1\}}||kr=ttt||ƒƒƒSq=Wt s„t dƒ‚dS(so Internal sample method Returns dictionary mapping RandomSymbol to realization value. Rziis)We should never have gotten to this pointN( R R^R|R2trandomtuniformR!RRtzipRIRl(RRoRvRVtvalueRu((s.lib/python2.7/site-packages/sympy/stats/frv.pytsampleCs N(R"R#R$R2R3R6RiRmRrRRwRIR|R}R~tNoneRRƒR„R‰(((s.lib/python2.7/site-packages/sympy/stats/frv.pyRbës        tSingleFinitePSpacecB s2eZdZed„ƒZeed„ƒƒZRS(s A single finite probability space Represents the probabilities of a set of random events that can be attributed to a single variable/symbol. This class is implemented by many of the standard FiniteRV types such as Die, Bernoulli, Coin, etc.... cC st|j|jjƒS(N(R4R8t distributionR9(R((s.lib/python2.7/site-packages/sympy/stats/frv.pyREasc s&t‡fd†ˆjjjƒDƒƒS(Nc3 s0|]&\}}tˆj|fƒ|fVqdS(N(RR8(R'R)Rq(R(s.lib/python2.7/site-packages/sympy/stats/frv.pys hs(R!RŒR>(R((Rs.lib/python2.7/site-packages/sympy/stats/frv.pyRdes(R"R#R$R%RERRd(((s.lib/python2.7/site-packages/sympy/stats/frv.pyR‹Ws tProductFinitePSpacecB sYeZdZed„ƒZeed„ƒƒZeed„ƒƒZd„Zd„Z RS(sG A collection of several independent finite probability spaces cC s#tg|jD]}|j^q ŒS(N(R=tspacesRE(Rtspace((s.lib/python2.7/site-packages/sympy/stats/frv.pyREpsc C sœtg|jD]}t|jjƒƒ^q Œ}i}xX|D]P}tt|Œƒ\}}t|ƒ}t|Œ}|j |dƒ|||Wt |ƒS(Ni( RRŽtiterRdR>RRR‡RRRhR( RRR@RpR>telemstprobsR<Rq((s.lib/python2.7/site-packages/sympy/stats/frv.pyRdts+   cC s t|jƒS(N(RRd(R((s.lib/python2.7/site-packages/sympy/stats/frv.pyRescC stj||ƒS(N(RbRƒ(RRF((s.lib/python2.7/site-packages/sympy/stats/frv.pyRƒ†scC stj||ƒS(N(RbRr(RRo((s.lib/python2.7/site-packages/sympy/stats/frv.pyRr‰s( R"R#R$R%RERRdReRƒRr(((s.lib/python2.7/site-packages/sympy/stats/frv.pyRls  (/R$t __future__RRt itertoolsRtsympyRRRRRRR R R R R RRRtsympy.sets.setsRtsympy.stats.rvRRRRRRRRRRtsympy.core.containersRR…R!RR&R4R=RARQRbR‹R(((s.lib/python2.7/site-packages/sympy/stats/frv.pyt s ^F %:.l