ó ”¼™\c@ s2dZddlmZmZddlmZmZmZmZm Z m Z m Z m Z m Z mZmZmZmZmZmZmZmZddlmZddlmZddlmZddlmZddlmZm Z m!Z!dd l"m#Z#d efd „ƒYZ$d e$fd „ƒYZ%de$fd„ƒYZ&defd„ƒYZ'de'fd„ƒYZ(defd„ƒYZ)de'fd„ƒYZ*de*fd„ƒYZ+de$fd„ƒYZ,d„Z-d„Z.d„Z/d„Z0d9d „Z2d9d9e3d!„Z4d9d9e3d"„Z5d#efd$„ƒYZ6d9e3d9d%„Z7d9e3d&„Z8d9e3d'„Z9d9e3d(„Z:d9d)„Z;d9d*„Z<d9ej=d+„Z>d9ej=d,„Z?d9ej=d-„Z@d9d.e3d/„ZAd9d.e3d0„ZBd9d.d1„ZCd2„ZDd3„ZEd4„ZFd9d5„ZGd6eHfd7„ƒYZId8„ZJd9S(:s  Main Random Variables Module Defines abstract random variable type. Contains interfaces for probability space object (PSpace) as well as standard operators, P, E, sample, density, where See Also ======== sympy.stats.crv sympy.stats.frv sympy.stats.rv_interface i’’’’(tprint_functiontdivision(tBasictStExprtSymboltTupletAndtAddtEqtlambdifytEqualitytLambdatsympifytDummytNetKroneckerDeltat DiracDeltatMul(tx(t string_types(t Relational(tBoolean(t FiniteSett ProductSett Intersection(tsolvesett RandomDomaincB s_eZdZeZeZeZeZd„Ze d„ƒZ e d„ƒZ d„Z d„Z RS(s¬ Represents a set of variables and the values which they can take See Also ======== sympy.stats.crv.ContinuousDomain sympy.stats.frv.FiniteDomain cG st|Œ}tj|||ŒS(N(RRt__new__(tclstsymbolstargs((s-lib/python2.7/site-packages/sympy/stats/rv.pyR-s cC s |jdS(Ni(R(tself((s-lib/python2.7/site-packages/sympy/stats/rv.pyR1scC s |jdS(Ni(R(R ((s-lib/python2.7/site-packages/sympy/stats/rv.pytset5scC s tƒ‚dS(N(tNotImplementedError(R tother((s-lib/python2.7/site-packages/sympy/stats/rv.pyt __contains__9scC s tƒ‚dS(N(R"(R texpr((s-lib/python2.7/site-packages/sympy/stats/rv.pytcompute_expectation<s(t__name__t __module__t__doc__tFalsetis_ProductDomaint is_Finitet is_Continuoust is_DiscreteRtpropertyRR!R$R&(((s-lib/python2.7/site-packages/sympy/stats/rv.pyRs   t SingleDomaincB s>eZdZd„Zed„ƒZed„ƒZd„ZRS(s˜ A single variable and its domain See Also ======== sympy.stats.crv.SingleContinuousDomain sympy.stats.frv.SingleFiniteDomain cC s"|jst‚tj|||ƒS(N(t is_SymboltAssertionErrorRR(RtsymbolR!((s-lib/python2.7/site-packages/sympy/stats/rv.pyRJscC s |jdS(Ni(R(R ((s-lib/python2.7/site-packages/sympy/stats/rv.pyR3NscC s t|jƒS(N(RR3(R ((s-lib/python2.7/site-packages/sympy/stats/rv.pyRRscC sHt|ƒdkrtSt|ƒd\}}|j|koG||jkS(Nii(tlenR*ttupleR3R!(R R#tsymtval((s-lib/python2.7/site-packages/sympy/stats/rv.pyR$Vs(R'R(R)RR/R3RR$(((s-lib/python2.7/site-packages/sympy/stats/rv.pyR0@s   tConditionalDomaincB s\eZdZd„Zed„ƒZed„ƒZed„ƒZed„ƒZd„Z RS(s« A RandomDomain with an attached condition See Also ======== sympy.stats.crv.ConditionalContinuousDomain sympy.stats.frv.ConditionalFiniteDomain cC s8|jtd„t|ƒDƒƒƒ}tj|||ƒS(Ncs s|]}||jfVqdS(N(R3(t.0trs((s-lib/python2.7/site-packages/sympy/stats/rv.pys hs(txreplacetdicttrandom_symbolsRR(Rt fulldomaint condition((s-lib/python2.7/site-packages/sympy/stats/rv.pyRgscC s |jjS(N(R>R(R ((s-lib/python2.7/site-packages/sympy/stats/rv.pyRlscC s |jdS(Ni(R(R ((s-lib/python2.7/site-packages/sympy/stats/rv.pyR>pscC s |jdS(Ni(R(R ((s-lib/python2.7/site-packages/sympy/stats/rv.pyR?tscC stdƒ‚dS(Ns)Set of Conditional Domain not Implemented(R"(R ((s-lib/python2.7/site-packages/sympy/stats/rv.pyR!xscC st|jjƒ|jƒS(N(RR>t as_booleanR?(R ((s-lib/python2.7/site-packages/sympy/stats/rv.pyR@|s( R'R(R)RR/RR>R?R!R@(((s-lib/python2.7/site-packages/sympy/stats/rv.pyR8]s  tPSpacecB seZdZd Zd Zd Zd Zed„ƒZ ed„ƒZ ed„ƒZ ed„ƒZ d„Z d„Zd„Zd„Zd „ZRS( s[ A Probability Space Probability Spaces encode processes that equal different values probabilistically. These underly Random Symbols which occur in SymPy expressions and contain the mechanics to evaluate statistical statements. See Also ======== sympy.stats.crv.ContinuousPSpace sympy.stats.frv.FinitePSpace cC s |jdS(Ni(R(R ((s-lib/python2.7/site-packages/sympy/stats/rv.pytdomain”scC s |jdS(Ni(R(R ((s-lib/python2.7/site-packages/sympy/stats/rv.pytdensity˜sc st‡fd†ˆjDƒƒS(Nc3 s|]}t|ˆƒVqdS(N(t RandomSymbol(R9R6(R (s-lib/python2.7/site-packages/sympy/stats/rv.pys žs(t frozensetR(R ((R s-lib/python2.7/site-packages/sympy/stats/rv.pytvaluesœscC s |jjS(N(RBR(R ((s-lib/python2.7/site-packages/sympy/stats/rv.pyR scC s tƒ‚dS(N(R"(R R?((s-lib/python2.7/site-packages/sympy/stats/rv.pytwhere¤scC s tƒ‚dS(N(R"(R R%((s-lib/python2.7/site-packages/sympy/stats/rv.pytcompute_density§scC s tƒ‚dS(N(R"(R ((s-lib/python2.7/site-packages/sympy/stats/rv.pytsampleŖscC s tƒ‚dS(N(R"(R R?((s-lib/python2.7/site-packages/sympy/stats/rv.pyt probability­scC s tƒ‚dS(N(R"(R R%((s-lib/python2.7/site-packages/sympy/stats/rv.pyR&°sN(R'R(R)tNoneR,R-R.tis_realR/RBRCRFRRGRHRIRJR&(((s-lib/python2.7/site-packages/sympy/stats/rv.pyRA€s     t SinglePSpacecB sSeZdZd„Zed„ƒZed„ƒZed„ƒZed„ƒZRS(sx Represents the probabilities of a set of random events that can be attributed to a single variable/symbol. cC sOt|tƒrt|ƒ}nt|tƒs<tdƒ‚ntj|||ƒS(Ns#s should have been string or Symbol(t isinstanceRRt TypeErrorRR(Rtst distribution((s-lib/python2.7/site-packages/sympy/stats/rv.pyR¹s cC st|j|ƒS(N(RDR3(R ((s-lib/python2.7/site-packages/sympy/stats/rv.pytvalueĄscC s |jdS(Ni(R(R ((s-lib/python2.7/site-packages/sympy/stats/rv.pyR3ÄscC s |jdS(Ni(R(R ((s-lib/python2.7/site-packages/sympy/stats/rv.pyRQČscC s|jj|jƒS(N(RQtpdfR3(R ((s-lib/python2.7/site-packages/sympy/stats/rv.pyRSĢs( R'R(R)RR/RRR3RQRS(((s-lib/python2.7/site-packages/sympy/stats/rv.pyRM“s  RDcB s”eZdZd d„ZeZeZeZeZ e d„ƒZ e d„ƒZ e d„ƒZ d„Zd„Zd„Ze d„ƒZd „Ze d „ƒZRS( s] Random Symbols represent ProbabilitySpaces in SymPy Expressions In principle they can take on any value that their symbol can take on within the associated PSpace with probability determined by the PSpace Density. Random Symbols contain pspace and symbol properties. The pspace property points to the represented Probability Space The symbol is a standard SymPy Symbol that is used in that probability space for example in defining a density. You can form normal SymPy expressions using RandomSymbols and operate on those expressions with the Functions E - Expectation of a random expression P - Probability of a condition density - Probability Density of an expression given - A new random expression (with new random symbols) given a condition An object of the RandomSymbol type should almost never be created by the user. They tend to be created instead by the PSpace class's value method. Traditionally a user doesn't even do this but instead calls one of the convenience functions Normal, Exponential, Coin, Die, FiniteRV, etc.... cC s›ddlm}|dkr(tƒ}nt|tƒsFtdƒ‚nt|tƒsdtdƒ‚n||krˆt|tƒrˆt}nt j |||ƒS(Ni’’’’(tJointRandomSymbolssymbol should be of type Symbols(pspace variable should be of type PSpace( tsympy.stats.joint_rvRTRKRARNRRORMRDRR(RR3tpspaceRT((s-lib/python2.7/site-packages/sympy/stats/rv.pyRės   cC s |jdS(Ni(R(R ((s-lib/python2.7/site-packages/sympy/stats/rv.pytžtcC s |jdS(Ni(R(R ((s-lib/python2.7/site-packages/sympy/stats/rv.pyRW’RXcC s |jjS(N(R3tname(R ((s-lib/python2.7/site-packages/sympy/stats/rv.pyRWRXcC s |jjS(N(R3t is_positive(R ((s-lib/python2.7/site-packages/sympy/stats/rv.pyt_eval_is_positivescC s |jjS(N(R3t is_integer(R ((s-lib/python2.7/site-packages/sympy/stats/rv.pyt_eval_is_integerscC s|jjp|jjS(N(R3RLRV(R ((s-lib/python2.7/site-packages/sympy/stats/rv.pyt _eval_is_realscC s |jjS(N(R3tis_commutative(R ((s-lib/python2.7/site-packages/sympy/stats/rv.pyR_ scC s|j|jfS(N(RVR3(R ((s-lib/python2.7/site-packages/sympy/stats/rv.pyt_hashable_contentscC s|hS(N((R ((s-lib/python2.7/site-packages/sympy/stats/rv.pyt free_symbolssN(R'R(R)RKRtTruet is_finitet is_symboltis_Atomt _diff_wrtR/RVR3RYR[R]R^R_R`Ra(((s-lib/python2.7/site-packages/sympy/stats/rv.pyRDŃs     t ProductPSpacecB seZdZRS(sĶ Abstract class for representing probability spaces with multiple random variables. See Also ======== sympy.stats.rv.IndependentProductPSpace sympy.stats.joint_rv.JointPSpace (R'R(R)(((s-lib/python2.7/site-packages/sympy/stats/rv.pyRgs tIndependentProductPSpacecB sæeZdZd„Zed„ƒZed„ƒZed„ƒZed„ƒZed„ƒZ de d„Z ed„ƒZ ed „ƒZd „Zd „Zd „Zd „Zed„ZRS(s A probability space resulting from the merger of two independent probability spaces. Often created using the function, pspace c  sųi}x,|D]$}x|jD]}|||6s sOverlapping Random Variablescs s|]}|jVqdS(N(R,(R9Rk((s-lib/python2.7/site-packages/sympy/stats/rv.pys ;s(tProductFinitePSpace(RFRtkeysR3RURiRjR4tsumt ValueErrortalltsympy.stats.frvRlRR( Rtspacest rs_space_dictRkRRR7RRltobj((RjRis-lib/python2.7/site-packages/sympy/stats/rv.pyR,s (+ cC sEtg|jD]}|j^q Œ}|jtd„|jDƒƒƒS(Ncs s|]}||jfVqdS(N(R3(R9trv((s-lib/python2.7/site-packages/sympy/stats/rv.pys Fs(RRrRStsubsR<RF(R Rktp((s-lib/python2.7/site-packages/sympy/stats/rv.pyRSCs%cC s<i}x/|jD]$}x|jD]}|||Zs(tsumsetsRr(R ((s-lib/python2.7/site-packages/sympy/stats/rv.pyRFXscK sw|p |j}t|ƒ}x3|jD](}|j|||j@dt|}q%W|rst|dƒrs|j|S|S(Ntevaluatetdoit(RFRERrR&R*thasattrR{(R R%trvsRztkwargsRk((s-lib/python2.7/site-packages/sympy/stats/rv.pyR&\s & cC s#tg|jD]}|j^q ŒS(N(t ProductDomainRrRB(R Rk((s-lib/python2.7/site-packages/sympy/stats/rv.pyRBescC stdƒ‚dS(Ns'Density not available for ProductSpaces(R"(R ((s-lib/python2.7/site-packages/sympy/stats/rv.pyRCiscC sd„|jDƒS(NcS s8i|].}|jƒjƒD]\}}||“qqS((RItitems(R9Rktktv((s-lib/python2.7/site-packages/sympy/stats/rv.pys ns (Rr(R ((s-lib/python2.7/site-packages/sympy/stats/rv.pyRImscK sĪt}t|tƒr;t|jd|jdƒ}t}n|j|j}t|ƒ}t ddtdtƒ}|j |ƒ}t g|D]}t |ƒj ^qˆƒrjddlm} m} ||jkr.tt|jƒt|gƒƒ} td„| Dƒƒ} |jj|j| |} t|j| ƒS| |ƒ}| ||ƒ}|j|j|jdƒƒ}nOdd lm}m}||ƒ}|||ƒ}|j|j|jdƒƒ}|sĆ|St j!|S( NiitztrealtFinitei’’’’(tContinuousDistributionHandmadetSingleContinuousPSpacecs s|]}|jVqdS(N(R3(R9R:((s-lib/python2.7/site-packages/sympy/stats/rv.pys €s(tDiscreteDistributionHandmadetSingleDiscretePSpace("R*RNRR RRbtlhstrhsR=RRHtanyRVR-tsympy.stats.crvR†R‡RFR5R!RERBt integrateRSR R3RJt __class__RRtsympy.stats.drvRˆR‰RtOne(R R?R~tcond_invR%R}RƒtdensRuR†R‡t randomsymbolsRRSRktresultRˆR‰((s-lib/python2.7/site-packages/sympy/stats/rv.pyRJqs.  (" ! cK stddtdtƒ}t|ƒ}td„|DƒƒrY|jt||ƒ|}n|jt||ƒ|}t||ƒS(NRƒR„tfinitecs s|]}t|ƒjVqdS(N(RVR-(R9Ru((s-lib/python2.7/site-packages/sympy/stats/rv.pys ‘s(RRbR=RŒR&RRR (R R%R~RƒR}((s-lib/python2.7/site-packages/sympy/stats/rv.pyRHŽs   cK stdƒ‚dS(Ns0CDF not well defined on multivariate expressions(Ro(R R%R~((s-lib/python2.7/site-packages/sympy/stats/rv.pyt compute_cdf™scK sŽt|ƒ}|jtd„|jDƒƒƒ}tg|D]}t|ƒj^q8ƒr‡ddlm}m }|}||j |ƒ} n”tg|D]}t|ƒj ^q‘ƒrąddl m } m} | }| |j |ƒ} nHtg|D]}t|ƒj^qźƒr(ddlm} | j||ƒS|rd„|jDƒ} | j|j|}|j|j| ƒ}t| j|ƒ}n|| |ƒS(Ncs s|]}||jfVqdS(N(R3(R9Ru((s-lib/python2.7/site-packages/sympy/stats/rv.pys žsi’’’’(tConditionalContinuousDomaintContinuousPSpace(tConditionalDiscreteDomaintDiscretePSpace(t FinitePSpacecS s%i|]}tt|ƒƒ|“qS((Rtstr(R9Ru((s-lib/python2.7/site-packages/sympy/stats/rv.pys ­s (R=R;R<RFRŒRVR-RR˜R™RBR.RRšR›RpR,RqRœtconditional_spaceRR&RSR (R R?t normalizeR~R}RuR˜R™RkRBRšR›Rœt replacementtnormRSRC((s-lib/python2.7/site-packages/sympy/stats/rv.pyRžœs& "(((N(R'R(R)RR/RSRsRRrRFRKR*R&RBRCRIRJRHR—RbRž(((s-lib/python2.7/site-packages/sympy/stats/rv.pyRh$s     RcB skeZdZeZd„Zed„ƒZed„ƒZed„ƒZ ed„ƒZ d„Z d„Z RS(s¶ A domain resulting from the merger of two independent domains See Also ======== sympy.stats.crv.ProductContinuousDomain sympy.stats.frv.ProductFiniteDomain cG ség}x7|D]/}|js,|j|ƒq |j|jƒq Wt|Œ}td„|Dƒƒr{ddlm}|}ntd„|DƒƒrŖddlm }|}ntd„|DƒƒrŁddl m }|}nt j ||ŒS(Ncs s|]}|jVqdS(N(R,(R9RB((s-lib/python2.7/site-packages/sympy/stats/rv.pys Ési’’’’(tProductFiniteDomaincs s|]}|jVqdS(N(R-(R9RB((s-lib/python2.7/site-packages/sympy/stats/rv.pys Ģs(tProductContinuousDomaincs s|]}|jVqdS(N(R.(R9RB((s-lib/python2.7/site-packages/sympy/stats/rv.pys Ļs(tProductDiscreteDomain(R+tappendtextendtdomainsRRpRqR¢RR£RR¤RR(RR§tdomains2RBR¢R£R¤((s-lib/python2.7/site-packages/sympy/stats/rv.pyRæs       cC std„|jDƒƒS(Ncs s+|]!}|jD]}||fVqqdS(N(R(R9RBR3((s-lib/python2.7/site-packages/sympy/stats/rv.pys ×s(R<R§(R ((s-lib/python2.7/site-packages/sympy/stats/rv.pytsym_domain_dictÕscC s0tg|jD]}|jD] }|^qq ŒS(N(RR§R(R RBR6((s-lib/python2.7/site-packages/sympy/stats/rv.pyRŚscC s|jS(N(R(R ((s-lib/python2.7/site-packages/sympy/stats/rv.pyR§ßscC std„|jDƒƒS(Ncs s|]}|jVqdS(N(R!(R9RB((s-lib/python2.7/site-packages/sympy/stats/rv.pys ås(RR§(R ((s-lib/python2.7/site-packages/sympy/stats/rv.pyR!ćscC slxe|jD]Z}tg|D]1}t|jj|dƒƒtjkr|^qƒ}||kr tSq WtS(Ni( R§RER RtcontainsRttrueR*Rb(R R#RBtitemtelem((s-lib/python2.7/site-packages/sympy/stats/rv.pyR$ēs cC s&tg|jD]}|jƒ^q ŒS(N(RR§R@(R RB((s-lib/python2.7/site-packages/sympy/stats/rv.pyR@ōs( R'R(R)RbR+RR/R©RR§R!R$R@(((s-lib/python2.7/site-packages/sympy/stats/rv.pyR“s  cC s6t|ddƒ}|dk r.t|tƒƒSgSdS(s> Returns all RandomSymbols within a SymPy Expression. tatomsN(tgetattrRKtlistRD(R%R®((s-lib/python2.7/site-packages/sympy/stats/rv.pyR=ųs c st|ƒ}t|tƒr1|jdk r1|jSt|ƒ‰ˆsVtd|ƒ‚nt‡fd†ˆDƒƒr}ˆdjStgˆD]}|j^q‡ŒS(s3 Returns the underlying Probability Space of a random expression. For internal use. Examples ======== >>> from sympy.stats import pspace, Normal >>> from sympy.stats.rv import IndependentProductPSpace >>> X = Normal('X', 0, 1) >>> pspace(2*X + 1) == X.pspace True s6Expression containing Random Variable expected, not %sc3 s%|]}|jˆdjkVqdS(iN(RV(R9Ru(R}(s-lib/python2.7/site-packages/sympy/stats/rv.pys siN( R RNRDRVRKR=RoRpRh(R%Ru((R}s-lib/python2.7/site-packages/sympy/stats/rv.pyRVs   cC stƒj|ŒS(s Union of sets (REtunion(tsets((s-lib/python2.7/site-packages/sympy/stats/rv.pyRyscC sNi}xA|D]9}g|D]}|j|jkr|^qd||>> from sympy.stats import given, density, Die >>> X = Die('X', 6) >>> Y = given(X, X > 3) >>> density(Y).dict {4: 1/3, 5: 1/3, 6: 1/3} Following convention, if the condition is a random symbol then that symbol is considered fixed. >>> from sympy.stats import Normal >>> from sympy import pprint >>> from sympy.abc import z >>> X = Normal('X', 0, 1) >>> Y = Normal('Y', 0, 1) >>> pprint(density(X + Y, Y)(z), use_unicode=False) 2 -(-Y + z) ----------- ___ 2 \/ 2 *e ------------------ ____ 2*\/ pi ii(R=tpspace_independentRNRDR R3R R4RVRBR8R5RRRtRealsRR°RvRbR*RRžR·RFR;( R%R?R~t condsymbolsRutresultstsumstresttempt fullspaceRktswapdict((s-lib/python2.7/site-packages/sympy/stats/rv.pytgiven9s4# !!    cK sĖt|ƒs|S|r)t||d|ƒS|dk rNtt||ƒd|ƒS|jrƒtg|jD]}t|d|ƒ^qdŒSt|ƒj |d||}|rĆt |dƒrĆ|j |S|SdS(sL Returns the expected value of a random expression Parameters ========== expr : Expr containing RandomSymbols The expression of which you want to compute the expectation value given : Expr containing RandomSymbols A conditional expression. E(X, X>0) is expectation of X given X > 0 numsamples : int Enables sampling and approximates the expectation with this many samples evalf : Bool (defaults to True) If sampling return a number rather than a complex expression evaluate : Bool (defaults to True) In case of continuous systems return unevaluated integral Examples ======== >>> from sympy.stats import E, Die >>> X = Die('X', 6) >>> E(X) 7/2 >>> E(2*X + 1) 8 >>> E(X, X > 3) # Expectation of X given that it is above 3 5 t numsamplesRzR{N( R=t sampling_ERKt expectationRĮtis_AddRRRVR&R|R{(R%R?RĀRzR~targR•((s-lib/python2.7/site-packages/sympy/stats/rv.pyRĂs   & cK s4t|ƒ}t|ƒ}|dk rMt|ttfƒ rMtd|ƒ‚n|tkr`tjSt|ttfƒsˆtd|ƒ‚n|tj kržtj S|tj kr“tjS|rŠt ||d||S|dk rõt t||||St|ƒj ||}|r,t|dƒr,|jƒS|SdS(sz Probability that a condition is true, optionally given a second condition Parameters ========== condition : Combination of Relationals containing RandomSymbols The condition of which you want to compute the probability given_condition : Combination of Relationals containing RandomSymbols A conditional expression. P(X > 1, X > 0) is expectation of X > 1 given X > 0 numsamples : int Enables sampling and approximates the probability with this many samples evaluate : Bool (defaults to True) In case of continuous systems return unevaluated integral Examples ======== >>> from sympy.stats import P, Die >>> from sympy import Eq >>> X, Y = Die('X', 6), Die('Y', 6) >>> P(X > 3) 1/2 >>> P(Eq(X, 5), X > 2) # Probability that X == 5 given that X > 2 1/4 >>> P(X > Y) 5/12 s4%s is not a relational or combination of relationalsRĀR{N(R RKRNRRRoR*RtZeroR«R‘tfalset sampling_PRJRĮRVR|R{(R?tgiven_conditionRĀRzR~R•((s-lib/python2.7/site-packages/sympy/stats/rv.pyRJ¹s0        tDensitycB s2eZed„ƒZed„ƒZed„ZRS(cC s |jdS(Ni(R(R ((s-lib/python2.7/site-packages/sympy/stats/rv.pyRWśRXcC s(t|jƒdkr |jdSdSdS(Ni(R4RRK(R ((s-lib/python2.7/site-packages/sympy/stats/rv.pyR?üs cK sddlm}|j|j}}|dk rDt|||}nt|tƒrot|j|ƒro|jj St |ƒs’t t t t |ƒƒSt|tƒrŅt|jdƒrŅtt|ƒtƒrŅ|jj St|ƒj||}|r t|dƒr |jƒS|SdS(Ni’’’’(t JointPSpaceRQR{(RURĢR%R?RKRĮRNRDRVRQR=R RRR|RMRHR{(R RzR~RĢR%R?R•((s-lib/python2.7/site-packages/sympy/stats/rv.pyR{s"     (R'R(R/R%R?RbR{(((s-lib/python2.7/site-packages/sympy/stats/rv.pyRĖłscK s8|rt||d||St||ƒjd||S(s' Probability density of a random expression, optionally given a second condition. This density will take on different forms for different types of probability spaces. Discrete variables produce Dicts. Continuous variables produce Lambdas. Parameters ========== expr : Expr containing RandomSymbols The expression of which you want to compute the density value condition : Relational containing RandomSymbols A conditional expression. density(X > 1, X > 0) is density of X > 1 given X > 0 numsamples : int Enables sampling and approximates the density with this many samples Examples ======== >>> from sympy.stats import density, Die, Normal >>> from sympy import Symbol >>> x = Symbol('x') >>> D = Die('D', 6) >>> X = Normal(x, 0, 1) >>> density(D).dict {1: 1/6, 2: 1/6, 3: 1/6, 4: 1/6, 5: 1/6, 6: 1/6} >>> density(2*D).dict {2: 1/6, 4: 1/6, 6: 1/6, 8: 1/6, 10: 1/6, 12: 1/6} >>> density(X)(x) sqrt(2)*exp(-x**2/2)/(2*sqrt(pi)) RĀRz(tsampling_densityRĖR{(R%R?RzRĀR~((s-lib/python2.7/site-packages/sympy/stats/rv.pyRCs&cK sd|dk r%tt||||St|ƒj||}|r\t|dƒr\|jƒS|SdS(s® Cumulative Distribution Function of a random expression. optionally given a second condition This density will take on different forms for different types of probability spaces. Discrete variables produce Dicts. Continuous variables produce Lambdas. Examples ======== >>> from sympy.stats import density, Die, Normal, cdf >>> D = Die('D', 6) >>> X = Normal('X', 0, 1) >>> density(D).dict {1: 1/6, 2: 1/6, 3: 1/6, 4: 1/6, 5: 1/6, 6: 1/6} >>> cdf(D) {1: 1/6, 2: 1/3, 3: 1/2, 4: 2/3, 5: 5/6, 6: 1} >>> cdf(3*D, D > 2) {9: 1/4, 12: 1/2, 15: 3/4, 18: 1} >>> cdf(X) Lambda(_z, erf(sqrt(2)*_z/2)/2 + 1/2) R{N(RKtcdfRĮRVR—R|R{(R%R?RzR~R•((s-lib/python2.7/site-packages/sympy/stats/rv.pyRĪGs   cK sd|dk r%tt||||St|ƒj||}|r\t|dƒr\|jƒS|SdS(s: Characteristic function of a random expression, optionally given a second condition Returns a Lambda Examples ======== >>> from sympy.stats import Normal, DiscreteUniform, Poisson, characteristic_function >>> X = Normal('X', 0, 1) >>> characteristic_function(X) Lambda(_t, exp(-_t**2/2)) >>> Y = DiscreteUniform('Y', [1, 2, 7]) >>> characteristic_function(Y) Lambda(_t, exp(7*_t*I)/3 + exp(2*_t*I)/3 + exp(_t*I)/3) >>> Z = Poisson('Z', 2) >>> characteristic_function(Z) Lambda(_t, exp(2*exp(_t*I) - 2)) R{N(RKtcharacteristic_functionRĮRVtcompute_characteristic_functionR|R{(R%R?RzR~R•((s-lib/python2.7/site-packages/sympy/stats/rv.pyRĻqs   cK sd|dk r%tt||||St|ƒj||}|r\t|dƒr\|jƒS|SdS(NR{(RKtmoment_generating_functionRĮRVt"compute_moment_generating_functionR|R{(R%R?RzR~R•((s-lib/python2.7/site-packages/sympy/stats/rv.pyRђs   cK s;|dk r%tt||||St|ƒj||S(sł Returns the domain where a condition is True. Examples ======== >>> from sympy.stats import where, Die, Normal >>> from sympy import symbols, And >>> D1, D2 = Die('a', 6), Die('b', 6) >>> a, b = D1.symbol, D2.symbol >>> X = Normal('x', 0, 1) >>> where(X**2<1) Domain: (-1 < x) & (x < 1) >>> where(X**2<1).set Interval.open(-1, 1) >>> where(And(D1<=D2 , D2<3)) Domain: (Eq(a, 1) & Eq(b, 1)) | (Eq(a, 1) & Eq(b, 2)) | (Eq(a, 2) & Eq(b, 2)) N(RKRGRĮRV(R?RŹR~((s-lib/python2.7/site-packages/sympy/stats/rv.pyRGs cK stt||ddƒƒS(sū A realization of the random expression Examples ======== >>> from sympy.stats import Die, sample >>> X, Y, Z = Die('X', 6), Die('Y', 6), Die('Z', 6) >>> die_roll = sample(X + Y + Z) # A random realization of three dice RĀi(tnextt sample_iter(R%R?R~((s-lib/python2.7/site-packages/sympy/stats/rv.pyRI¼s cK s?yt||||SWn!tk r:t||||SXdS(s„ Returns an iterator of realizations from the expression given a condition Parameters ========== expr: Expr Random expression to be realized condition: Expr, optional A conditional expression numsamples: integer, optional Length of the iterator (defaults to infinity) Examples ======== >>> from sympy.stats import Normal, sample_iter >>> X = Normal('X', 0, 1) >>> expr = X*X + 3 >>> iterator = sample_iter(expr, numsamples=3) >>> list(iterator) # doctest: +SKIP [12, 4, 7] See Also ======== Sample sampling_P sampling_E sample_iter_lambdify sample_iter_subs N(tsample_iter_lambdifyROtsample_iter_subs(R%R?RĀR~((s-lib/python2.7/site-packages/sympy/stats/rv.pyRŌĖs# c sõˆrtt|ˆƒƒ‰n t|ƒ‰tˆjƒ‰tˆ||‰ˆrftˆˆ|‰nyJˆjƒ}gˆD]}||^q|}ˆ|Œˆrƈ|ŒnWntk rĻtdƒ‚nX‡‡‡‡‡‡fd†}|ƒS(sd See sample_iter Uses lambdify for computation. This is fast but does not always work. s'Expr/condition too complex for lambdifyc3 s£d}x–|ˆkržˆjƒ}gˆD]}||^q(}ˆr†ˆ|Œ}|tkrw|tkrwtdƒ‚n|s†q q†nˆ|ŒV|d7}q WdS(Nis(Conditions must not contain free symbolsi(RIRbR*Ro(tcountRxRuRtgd(R?tfntgiven_fnRĀtpsR}(s-lib/python2.7/site-packages/sympy/stats/rv.pytreturn_generators     (RVRR°RFR RIt ExceptionRO(R%R?RĀR~RxRuRRÜ((R?RŁRŚRĀRŪR}s-lib/python2.7/site-packages/sympy/stats/rv.pyRÕõs"    ck sĀ|dk r$tt||ƒƒ}n t|ƒ}d}x…||kr½|jƒ}|dk r¢|j|ƒ}|tkr“|tkr“tdƒ‚n|s¢q9q¢n|j|ƒV|d7}q9WdS(s_ See sample_iter Uses subs for computation. This is slow but almost always works. is(Conditions must not contain free symbolsiN(RKRVRRIR;RbR*Ro(R%R?RĀR~RŪR×RxRŲ((s-lib/python2.7/site-packages/sympy/stats/rv.pyRÖ#s     ic K s”d}d}t||d||}xR|D]J}|tkrX|tkrXtdƒ‚n|rk|d7}q+|d7}q+Wt|ƒ|} |r™| jƒS| SdS(sf Sampling version of P See Also ======== P sampling_E sampling_density iRĀs(Conditions must not contain free symbolsiN(RŌRbR*RoRtevalf( R?RŹRĀRŽR~t count_truet count_falsetsamplesRIR•((s-lib/python2.7/site-packages/sympy/stats/rv.pyRÉ=s     cK sFt||d||}tt|ƒŒ|}|r>|jƒS|SdS(se Sampling version of E See Also ======== P sampling_P sampling_density RĀN(RŌRR°RŽ(R%RŹRĀRŽR~RįR•((s-lib/python2.7/site-packages/sympy/stats/rv.pyRĆas   cK sGi}x:t||d||D] }|j|dƒd||>> from sympy.stats import Normal, dependent, given >>> from sympy import Tuple, Eq >>> X, Y = Normal('X', 0, 1), Normal('Y', 0, 1) >>> dependent(X, Y) False >>> dependent(2*X + Y, -Y) True >>> X, Y = given(Tuple(X, Y), Eq(X + Y, 3)) >>> dependent(X, Y) True See Also ======== independent RƒR„(RøR*RRbRCR (R³R“Rƒ((s-lib/python2.7/site-packages/sympy/stats/rv.pyt dependent‰s $cC st||ƒ S(s Independence of two random expressions Two expressions are independent if knowledge of one does not change computations on the other. Examples ======== >>> from sympy.stats import Normal, independent, given >>> from sympy import Tuple, Eq >>> X, Y = Normal('X', 0, 1), Normal('Y', 0, 1) >>> independent(X, Y) True >>> independent(2*X + Y, -Y) False >>> X, Y = given(Tuple(X, Y), Eq(X + Y, 3)) >>> independent(X, Y) False See Also ======== dependent (Rć(R³R“((s-lib/python2.7/site-packages/sympy/stats/rv.pyt independent®scC s~tt|ƒjƒ}tt|ƒjƒ}ttt|ƒƒjt|ƒƒƒdkr[tSt|j|ƒƒdkrztSdS(sc Tests for independence between a and b by checking if their PSpaces have overlapping symbols. This is a sufficient but not necessary condition for independence and is intended to be used internally. Notes ===== pspace_independent(a, b) implies independent(a, b) independent(a, b) does not imply pspace_independent(a, b) iN( R!RVRR4R=t intersectionR*RbRK(R³R“t a_symbolst b_symbols((s-lib/python2.7/site-packages/sympy/stats/rv.pyRøĢs -cC sB|dkrt|ƒ}n|s%|Sd„|Dƒ}|j|ƒS(s£ Given a random expression replace all random variables with their symbols. If symbols keyword is given restrict the swap to only the symbols listed. cS si|]}|j|“qS((R3(R9Ru((s-lib/python2.7/site-packages/sympy/stats/rv.pys ķs N(RKR=R;(R%RRĄ((s-lib/python2.7/site-packages/sympy/stats/rv.pytrv_subsćs  tNamedArgsMixincB seZdZd„ZRS(cC sRy|j|jj|ƒSWn0tk rMtdt|ƒj|fƒ‚nXdS(Ns!'%s' object has no attribute '%s'(Rt _argnamestindexRotAttributeErrorttypeR'(R tattr((s-lib/python2.7/site-packages/sympy/stats/rv.pyt __getattr__ós  ((R'R(RźRļ(((s-lib/python2.7/site-packages/sympy/stats/rv.pyRéšscC s|tkrt|ƒ‚ndS(sh Check a condition on input value. Raises ValueError with message if condition is not True N(R*Ro(R?tmessage((s-lib/python2.7/site-packages/sympy/stats/rv.pyt _value_checkśs N(KR)t __future__RRtsympyRRRRRRRR R R R R RRRRRt sympy.abcRtsympy.core.compatibilityRtsympy.core.relationalRtsympy.logic.boolalgRtsympy.sets.setsRRRtsympy.solvers.solvesetRRR0R8RARMRDRgRhRR=RVRyR·RKRĮRbRÄRJRĖRCRĪRĻRŃRGRItInfinityRŌRÕRÖRÉRĆRĶRćRäRøRčtobjectRéRń(((s-lib/python2.7/site-packages/sympy/stats/rv.pyts\p##4F D    I7 ?!-*!  *. #  %