
    g$                         d dl mZ d dlmZ d dlmZmZ d dlmZ d dl	m
Z
mZ d dlmZ d dlmZ d dlmZ d d	lmZ d d
lmZ d dlmZmZmZmZmZmZmZmZmZ d dl m!Z!m"Z" d Z#d Z$d Z%d Z&dS )    )Sum)Mul)oopi)Eq)Dummysymbols)exp)sqrt)sin)Integral)	unchanged)	NormalPoissonvariance
CovarianceVarianceProbabilityExpectationMomentCentralMoment)probabilityexpectationc                  #   t          ddd          } t          ddd          }t          dd          }t          dd          }t          dd	
          }t          d          \  }}}t          | dk                                              t          | dk              k    sJ t          | |k                                              t          | |k              k    sJ t          | dk                                  t                                                    t          | dk              k    sJ t          | |k                                  t                                                    t          | |k              k    sJ t          |                                           t          |           k    sJ t          |                               t                                                    t          |           k    sJ t          | dz                                            t          | dz            k    sJ t          || z            j        || z  fk    sJ t          || z                                            |t          |           z  k    sJ t          d| z  d|z  z   || z  |z  z                                             dt          |           z  dt          |          z  z   |t          | |z            z  z   k    sJ t          d| z  d|z  z   || z  |z  z             j        d| z  d|z  z   || z  |z  z   fk    sJ t          t          |                     t          t          |                                                     k    sJ t          d|z  t          |           z  |z  || dz  z  z   || z  |z  z                                             d|z  t          t          |           |z            z  |t          | dz            z  z   |t          | |z            z  z   k    sJ t          | |z                                             t          |           t          |          z   k    sJ t          | |z   | |z
  z                                            t          | dz            t          |dz            z
  k    sJ t          | |z   | |z
  z                                                                            dk    sJ t          | |z   d	                                          dk    sJ t          | t          |          z                                             dk    sJ t          | t          |          z                                 d          dt          t          |                    z   k    sJ t          | t          |t          d| z            z             z                                 d          dt          t          |t          d| z            z                       z   k    sJ t          | t          |t          d| z            z             z                                             dk    sJ t          t          d| z                                                      dk    sJ t          t          d| z                                          d          t          d| z            k    sJ t          dt          d| z            z                                d          dt          d| z            z  k    sJ t          | |z   dz                                            dt          | |dz  z            z  dt          | dz  |z            z  z   t          | dz            z   t          |dz            z   k    sJ t          | |z
  dz                                            dt          | |dz  z            z  dt          | dz  |z            z  z
  t          | dz            z   t          |dz            z
  k    sJ t          | |z
  dz                                            dt          | |z            z  t          | dz            z   t          |dz            z   k    sJ t          |          j        |fk    sJ t          |                                          dk    sJ t          |                                           t          |                               t                                                    cxk    rt          |           k    sn J t          | |z             j        | |z   fk    sJ t          | |z                                             t          |           k    sJ t          | |z            j        t!          | |          fk    sJ t#          t          | |z                      t          k    sJ t          || z                                            |dz  t          |           z  k    sJ t          | |z                                             t          |           t          |          z   dt%          | |          z  z   k    sJ t          | |z   |z   |z                                             t          |           t          |          z   t          |          z   t          |          z   dt%          | |          z  z   dt%          | |          z  z   dt%          | |          z  z   dt%          ||          z  z   dt%          ||          z  z   dt%          ||          z  z   k    sJ t          | dz                                            t          | dz            k    sJ t'          t          | dz            sJ t          || dz  z                                            |dz  t          | dz            z  k    sJ t          t          |                     j        t          |           fk    sJ t          t          |                                                     t          t          |                     k    sJ t          |t          |           z                                            |dz  t          t          |                     z  k    sJ t%          ||          j        ||fk    sJ t%          ||                                          dk    sJ t%          | |                                          dk    sJ t%          ||                                           dk    sJ t%          | |          j        | |fk    sJ t#          t%          | |                    t$          k    sJ t%          || z  dz   |                                          |t%          | |          z  k    sJ t%          | |           j        | | fk    sJ t%          | |                                           t          |           k    sJ t%          || z  dz   ||z  dz                                             ||z  t%          | |          z  k    sJ t%          | |          t%          ||           k    sJ t%          | |z   ||z                                             t%          ||           t%          ||          z   t%          | |          z   t%          ||          z   k    sJ t%          || z  ||z  z   ||z  ||z  z                                             ||z  t%          ||           z  ||z  t%          ||          z  z   ||z  t%          | |          z  z   ||z  t%          ||          z  z   k    sJ t%          || dz  z  |t          |          z  z   ||z  |dz  z  ||z  z                                             ||z  t%          || dz            z  ||z  t%          t          |          |          z  z   ||z  t%          ||dz  z  | dz            z  z   ||z  t%          ||dz  z  t          |                    z  z   k    sJ t%          | | dz                                            t%          | | dz            k    sJ t%          | t          |                                                     t%          t          |           |           k    sJ t%          | dz  t          |           |z                                            t%          t          |           |z  | dz            k    sJ t%          ||                                           dk    sJ d S )NX      Y   ZWxTrealzy, w, zr   i)evaluate   F)deep	   )r   r   r	   r   evaluate_integralr   rewriter   doitr   r   argsexpandr   r   r   r   typer   r   )r   r   r    r!   r"   ywzs           g/var/www/html/ai-engine/env/lib/python3.11/site-packages/sympy/stats/tests/test_symbolic_probability.pytest_literal_probabilityr4      s\   sAqAsAqAQAQA$Ai  GAq!q1u//11[Q5G5GGGGGq1u//11[Q5G5GGGGGq1u%%h//4466+a!e:L:LLLLLq1u%%h//4466+a!e:L:LLLLLq>>++--Q????q>>!!(++0022k!nnDDDDq!t..00K14E4EEEEEqs QqSF****qs""$$+a..(88888qsQqSy1Q3q5())0022aA6F;WX>>IY6Y\]^ijklmjm^n^n\n6nnnnnqsQqSy1Q3q5()).1Q319qs1u3D2FFFFFs1vv+c!ff"5"5"<"<">">>>>>qs3q66z!|a1f,qs1u455<<>>!A#kRUVWRXRXYZRZF[F[B[AqD 1 11C245k!A#6F6F4FCG G G G Gq1u$$&&;q>>KNN+JJJJJAA''..00K14E4ETUWXTXHYHY4YYYYYAA''..0055773>>>>q1ut,,,1133q8888q;q>>)**//11Q6666q;q>>)**//U/;;q;{[\~~C^C^?^^^^^q;q;qs+;+;';<<<==BBBNNRS"-k!k!A#>N>N:N.O.O"P"PSQ Q Q Q Qq;q;qs+;+;';<<<==BBDDIIII{1Q3''((--//14444{1Q3''((--5-99[1=M=MMMMMqQqS)))**//U/;;qQqSAQAQ?QQQQQAz""))++qQq!tV1D1D/D+ad1f%%%0&(3AqD(9(90:<G1<M<M0N N N N NAz""))++qQq!tV1D1D/D+ad1f%%%0&(3AqD(9(90:<G1<M<M0N N N N NAz""))++r+ac2B2B/BAqD!!0"$/1$5$506 6 6 6 6 A;;t####A;;1$$$$A;;((**hqkk.A.A(.K.K.P.P.R.RaaaaV^_`VaVaaaaaaaAE??AE8++++AE??!!##x{{2222AaC==#a))----1(****AaC==!!QT(1++%55555AE??!!##x{{Xa[['@1ZPQSTEUEUCU'UUUUUAEAIM""))++hqkk0IHUVKK0WZbcdZeZe0e'(:a+;+;';1<>?*QPQBRBR>R1SUVYcdeghYiYiUi1j'(:a+;+;';1<>?*QPQBRBR>R1SUVYcdeghYiYiUi1j k k k k AqD>>++--!Q$????Xq!t$$$$$AadF""$$1Xad^^(;;;;;CFF SVVI----CFF""$$Q(8(88888Ac!ffH$$&&!Q$xA/?/?*?????a QF****a""$$))))a""$$))))a""$$))))a QF****
1a  !!Z////acAgq!!((**a
1a0@0@.@@@@@a QF****a""$$3333acAgqsQw''..00AaC
1Q4GGGGGaz!Q//////a!eQU##**,,
1a0@0@:aQRCSCS0SV`abdeVfVf0fistuwxiyiy0yyyyyacAaCi1qs++22441Z1=M=M9MPQRSPST^_`bcTdTdPd9d01!Jq!4D4D0D:EGHs:VWYZK[K[G[:\ ] ] ] ]a1fqQx'1QTAaC88??AAac*UVXY[\X\J]J]F]`abc`cdnorstououwxdydy`yFy89!JqAvqRSt<T<T8TGUWXYZWZ[efghiklhlflnqrsntnt[u[uWuGv w w w waA%%'':aA+>+>>>>>aQ  ''))ZA-B-BBBBBadCFF1H%%,,..*SVVAXq!t2L2LLLLLa--//1444444    c                     t          ddd          } t          ddd          }t          dd          }t          dd          }t          d          \  }}}}t          |                              t
                    d	k    sJ t          |                               t
                    t          | dz            t          |           dz  z
  k    sJ t          | |
                              t
                    t          | dz  |          t          | |          dz  z
  k    sJ t          | |          t          | dz            t          |           dz  z
  k    sJ t          | |z                                 t
                    t          | |z   dz            t          | |z             dz  z
  k    sJ t          | |z                                t
                    t          | dz  |dz  z            t          | |z            dz  z
  k    sJ t          ||                               t
                    | t          |           z  t          || z            z   k    sJ t          | |                              t
                    t          | |z            t          |           t          |          z  z
  k    sJ t          | ||
                              t
                    t          | |z  |          t          | |          t          ||          z  z
  k    sJ t          d          \  }}}t          t          | |                    }t          t          ||                    }	t          |                               t                    t          ||z  |t           t          f          k    sJ t          |                              t                    t          ||	z  |d	t          f          k    sJ t          |                               t                    t          |dz  |z  |t           t          f          t          ||z  |t           t          f          dz  z
  k    sJ t          |                              t                    t          |dz  |	z  |d	t          f          t          ||	z  |d	t          f          dz  z
  k    sJ t          ||                               t                    | t          |t          t          | |                    z  |t           t          f          z  t          ||z  t          t          | |                    z  |t           t          f          z   k    sJ t          |                               t                    t          |                               t                    k    sJ t          |                               t                    t          |                               t                    k    sJ t          ||                               t                    d	k    sJ t          ||                               t                    d	k    sJ t          | |
                              t                    t          |dz  t          t          | |          |          z  |t           t          f          t          |t          t          | |          |          z  |t           t          f          dz  z
  k    sJ d S )Nr   r   r   r   r   r    r!   z
x, y, w, zr   )	conditionzW, x, z)r   r   r	   r   r+   r   r   r   r   r   r   r   )
r   r   r    r!   r"   r0   r1   r2   pxpzs
             r3   test_probability_rewriter:   c   s   sAqAsAqAQAQA&&JAq!QA;;{++q0000A;;{++{16/B/B[QR^^WXEX/XXXXXA###++K88KQPQ<R<RU`abdeUfUfjkUk<kkkkkAq>>[A..Q1BBBBBBAE??"";//;A!|3L3L{[\_`[`OaOaefOf3fffffAE??"";//;qAvQ3O3OR]^_bc^cRdRdhiRi3iiiiia##K00QB{1~~4ETUVWTWHXHX4XXXXXa##K00K!4D4D{ST~~VabcVdVdGd4dddddaa(((00==QQRUTUAVAVYdefhiYjYjmxyz|}m~m~Y~A~~~~~i  GAq!	R1XX		B	R1XX		Bq>>!!+..(1R4!bS"2N2NNNNNq>>!!+..#adQ2J2G2GGGGGA;;{++x1R!bS"/N/NQYZ[\^Z^abegdgik`lQmQmopQp/pppppA;;{++s1a47Q2J/G/G#aPRdUVXY[]T^J_J_abJb/bbbbba##K002hqR1XX...RC===1[Y[\]_`YaYaMbMbIbefikhkmodp@q@qqr r r r A;;s##x{{':':8'D'DDDDDq>>!!#&&+a..*@*@*J*JJJJJa##C((A----a##H--2222A###++K88HQT+VXYZ\]V^V^`aJbJbEbefikhkmodp<q<q<DQ{SUVWYZS[S[]^G_G_E_bcfhehjlam<n<npq<q=r r r r r r rr5   c            
      J   t          dd          } t          dd          }t          d          }t          d| |          }t          |dd	          }|                    t                    t	          |d	z
  dz            k    sJ |                    t
                    t          |d	z
  dz  t          t          ||                    z  |t           t          f          k    sJ t          d
          }t          t          d	          |d	z
  dz  z  t          || z
  d	z   d	|d	z  z  z            z  d	t          t                    z  |z  z  |t           t          f          }|                    t                                        |          sJ |                                | dz  d| dz  z  z
  d| d	z  z  |d	z  z  z   d| d	z  z  z   d| z  |d	z  z  z
  d| z  z
  d|dz  z  z   d|d	z  z  z   dz   k    sJ t          d	d          }|                                dk    sJ d S )NmuTr#   sigmapositiver"   r   r   r   k   r                r&   )r	   r   r   r+   r   r   r   r   r   r   r   r
   r   dummy_eqr,   )r<   r=   r"   r   Mr@   expris          r3   test_symbolic_MomentrI      sY   	D	!	!	!BGd+++EAsBAq!QA99[!![!a%!%<%<<<<<99[!!Xq1uqjR1XX9N9N.N%&RL&2 &2 2 2 2 2c

AT!WWa!eaZ'a/. -%(
-$ )% )% %&'Rj&68:;bS"G GE99X''.....6688A"a%!BE'%(*::2q5b5>*,.rE245eQhJ?ACE1HMOQR S S S Sq!A6688tr5   c                  "   t          dd          } t          dd          }t          d          }t          d| |          }t          |d          }|                    t                    t	          |t	          |          z
  dz            k    sJ |                    t
                    t          |t          |t          d          z  |t           t          f          z
  dz  t          t          ||                    z  |t           t          f          k    sJ t          d	          }t          t          d
          |t          t          d
          |z  t          || z
  d
z   d
|d
z  z  z            z  d
t          t                    z  |z  z  |t           t          f          z
  dz  z  t          || z
  d
z   d
|d
z  z  z            z  d
t          t                    z  |z  z  |t           t          f          }|                    t                                        |          sJ |                                                                d|dz  z  k    sJ t!          dd          }|                                dk    sJ d S )Nr<   Tr#   r=   r>   r"   r   rB   r@   r      r&   i5  )r	   r   r   r+   r   r   r   r   r   r   r   r
   r   rF   r,   simplifyr   )r<   r=   r"   r   CMr@   rH   s          r3   test_symbolic_CentralMomentrN      s   	D	!	!	!BGd+++EAsBA	q!		B::k""k1{1~~3E2I&J&JJJJJ::k""hHQ{4?P?P=PRL5" 5" 1"$%0&&1"Q((&;&;0<>?"b\'K 'K K K K Kc

AT!WWa(47719S1
<; :%(
: 6 6 ,Rj.,023bS"#? #? ?ABC CCF
JI H%(
H D DRj.023bS"? ?E ::h((/////77992eQh;....	1B7799r5   N)'sympy.concrete.summationsr   sympy.core.mulr   sympy.core.numbersr   r   sympy.core.relationalr   sympy.core.symbolr   r	   &sympy.functions.elementary.exponentialr
   (sympy.functions.elementary.miscellaneousr   (sympy.functions.elementary.trigonometricr   sympy.integrals.integralsr   sympy.core.exprr   sympy.statsr   r   r   r   r   r   r   r   r   sympy.stats.rvr   r   r4   r:   rI   rN    r5   r3   <module>r\      s   ) ) ) ) ) )       ' ' ' ' ' ' ' ' $ $ $ $ $ $ . . . . . . . . 6 6 6 6 6 6 9 9 9 9 9 9 8 8 8 8 8 8 . . . . . . % % % % % %J J J J J J J J J J J J J J J J J J J J J J 3 3 3 3 3 3 3 3P5 P5 P5f&r &r &rR  &    r5   