
    \h$                         S SK Jr  S SKJr  S SKJrJr  S SKJr  S SK	J
r
Jr  S SKJr  S SKJr  S SKJr  S S	KJr  S S
KJr  S SKJrJrJrJrJrJrJrJrJr  S SK J!r!J"r"  S r#S r$S r%S r&g)    )Sum)Mul)oopi)Eq)Dummysymbols)exp)sqrt)sin)Integral)	unchanged)	NormalPoissonvariance
CovarianceVarianceProbabilityExpectationMomentCentralMoment)probabilityexpectationc                     [        SSS5      n [        SSS5      n[        SS5      n[        SS5      n[        SS	S
9n[        S5      u  pVn[        U S:  5      R	                  5       [        U S:  5      :X  d   e[        X:  5      R	                  5       [        X:  5      :X  d   e[        U S:  5      R                  [        5      R                  5       [        U S:  5      :X  d   e[        X:  5      R                  [        5      R                  5       [        X:  5      :X  d   e[        U 5      R	                  5       [        U 5      :X  d   e[        U 5      R                  [        5      R                  5       [        U 5      :X  d   e[        U S-  5      R	                  5       [        U S-  5      :X  d   e[        X@-  5      R                  X@-  4:X  d   e[        X@-  5      R                  5       U[        U 5      -  :X  d   e[        SU -  SU-  -   Xp-  U-  -   5      R                  5       S[        U 5      -  S[        U5      -  -   U[        X-  5      -  -   :X  d   e[        SU -  SU-  -   Xp-  U-  -   5      R                  SU -  SU-  -   Xp-  U-  -   4:X  d   e[        [        U 5      5      [        [        U 5      5      R                  5       :X  d   e[        SU-  [        U 5      -  U-  XPS-  -  -   Xp-  U-  -   5      R                  5       SU-  [        [        U 5      U-  5      -  U[        U S-  5      -  -   U[        X-  5      -  -   :X  d   e[        X-   5      R                  5       [        U 5      [        U5      -   :X  d   e[        X-   X-
  -  5      R                  5       [        U S-  5      [        US-  5      -
  :X  d   e[        X-   X-
  -  5      R                  5       R                  5       S:X  d   e[        X-   S	S9R                  5       S:X  d   e[        U [        U5      -   5      R                  5       S:X  d   e[        U [        U5      -   5      R                  SS9S[        [        U5      5      -   :X  d   e[        U [        U[        SU -  5      -   5      -   5      R                  SS9S[        [        U[        SU -  5      -   5      5      -   :X  d   e[        U [        U[        SU -  5      -   5      -   5      R                  5       S:X  d   e[        [        SU -  5      5      R                  5       S:X  d   e[        [        SU -  5      5      R                  SS9[        SU -  5      :X  d   e[        S[        SU -  5      -  5      R                  SS9S[        SU -  5      -  :X  d   e[        X-   S-  5      R                  5       S[        XS-  -  5      -  S[        U S-  U-  5      -  -   [        U S-  5      -   [        US-  5      -   :X  d   e[        X-
  S-  5      R                  5       S[        XS-  -  5      -  S[        U S-  U-  5      -  -
  [        U S-  5      -   [        US-  5      -
  :X  d   e[        X-
  S-  5      R                  5       S[        X-  5      -  [        U S-  5      -   [        US-  5      -   :X  d   e[        U5      R                  U4:X  d   e[        U5      R                  5       S:X  d   e[        U 5      R	                  5       [        U 5      R                  [        5      R                  5       s=:X  a  [        U 5      :X  d   e   e[        X-   5      R                  X-   4:X  d   e[        X-   5      R                  5       [        U 5      :X  d   e[        X-  5      R                  [!        X5      4:X  d   e[#        [        X-  5      5      [        :X  d   e[        Xp-  5      R                  5       US-  [        U 5      -  :X  d   e[        X-   5      R                  5       [        U 5      [        U5      -   S[%        X5      -  -   :X  d   e[        X-   U-   U-   5      R                  5       [        U 5      [        U5      -   [        U5      -   [        U5      -   S[%        X5      -  -   S[%        X5      -  -   S[%        X5      -  -   S[%        X5      -  -   S[%        X5      -  -   S[%        X25      -  -   :X  d   e[        U S-  5      R	                  5       [        U S-  5      :X  d   e['        [        U S-  5      (       d   e[        X@S-  -  5      R                  5       US-  [        U S-  5      -  :X  d   e[        [        U 5      5      R                  [        U 5      4:X  d   e[        [        U 5      5      R                  5       [        [        U 5      5      :X  d   e[        U[        U 5      -  5      R                  5       US-  [        [        U 5      5      -  :X  d   e[%        Xg5      R                  Xg4:X  d   e[%        Xg5      R                  5       S:X  d   e[%        X5      R                  5       S:X  d   e[%        X`5      R                  5       S:X  d   e[%        X5      R                  X4:X  d   e[#        [%        X5      5      [$        :X  d   e[%        Xp-  S-   U5      R                  5       U[%        X5      -  :X  d   e[%        X 5      R                  X 4:X  d   e[%        X 5      R                  5       [        U 5      :X  d   e[%        Xp-  S-   Xa-  S-   5      R                  5       Xg-  [%        X5      -  :X  d   e[%        X5      [%        X5      :X  d   e[%        X-   X#-   5      R                  5       [%        X05      [%        X15      -   [%        X5      -   [%        X5      -   :X  d   e[%        X@-  XQ-  -   Xr-  Xc-  -   5      R                  5       XF-  [%        X05      -  Xe-  [%        X15      -  -   XG-  [%        X5      -  -   XW-  [%        X5      -  -   :X  d   e[%        X@S-  -  U[        U5      -  -   Xq-  US-  -  Xc-  -   5      R                  5       Xd-  [%        X0S-  5      -  Xe-  [%        [        U5      U5      -  -   XG-  [%        XS-  -  U S-  5      -  -   XW-  [%        XS-  -  [        U5      5      -  -   :X  d   e[%        X S-  5      R                  5       [%        X S-  5      :X  d   e[%        U [        U 5      5      R                  5       [%        [        U 5      U 5      :X  d   e[%        U S-  [        U 5      U-  5      R                  5       [%        [        U 5      U-  U S-  5      :X  d   e[%        X`5      R	                  5       S:X  d   eg )NX      Y   ZWxTrealzy, w, zr   i)evaluate   F)deep	   )r   r   r	   r   evaluate_integralr   rewriter   doitr   r   argsexpandr   r   r   r   typer   r   )r   r   r    r!   r"   ywzs           c/var/www/auris/envauris/lib/python3.13/site-packages/sympy/stats/tests/test_symbolic_probability.pytest_literal_probabilityr4      sU   sAqAsAqAQAQA$Ai GA!q1u//1[Q5GGGGqu//1[5GGGGq1u%%h/446+a!e:LLLLqu%%h/446+ae:LLLLq>++-Q???q>!!(+002k!nDDDq!t..0K14EEEEqs  QSF***qs""$+a.(8888qsQqSy13q5()002aA6F;WX>IY6Y\]^ijkjm^n\n6nnnnqsQqSy13q5()..1Q319qs1u3D2FFFFs1v+c!f"5"<"<">>>>qs3q6z!|a1f,qs1u45<<>!A#kRUVWRXYZRZF[B[AqD 11C245k!#6F4FCG G G Gqu$$&;q>KN+JJJJ'..0K14ETUWXTXHY4YYYY'..05573>>>qut,113q888q;q>)*//1Q666q;q>)*//U/;q;{[\~C^?^^^^q;q;qs+;';<<=BBBNRS"-k!k!A#>N:N.O"PSQ Q Q Qq;q;qs+;';<<=BBDIII{1Q3'(--/1444{1Q3'(--5-9[1=MMMMqQqS))*//U/;qQqSAQ?QQQQz"))+qQ!tV1D/D+ad1f%%0&(3AqD(90:<G1<M0N N N Nz"))+qQ!tV1D/D+ad1f%%0&(3AqD(90:<G1<M0N N N Nz"))+r+ac2B/BAqD!0"$/1$506 6 6 6 A;t###A;1$$$A;((*hqk.A.A(.K.P.P.RaV^_`VaaaaaaAE?AE8+++AE?!!#x{222AC=#a)---(***AC=!QT(1+%5555AE?!!#x{Xa['@1ZPQEUCU'UUUUAEAIM"))+hqk0IHUVK0WZbcdZe0e'(:a+;';1<>?*QBR>R1SUVYcdeYiUi1j'(:a+;';1<>?*QBR>R1SUVYcdeYiUi1j k k k AqD>++-!Q$???Xq!t$$$$AdF""$1Xad^(;;;;CF  SVI---CF""$Q(8888Ac!fH$$&!Q$xA/?*????a  QF***a""$)))a""$)))a""$)))a  QF***
1 !Z///acAgq!((*a
10@.@@@@a  QF***a""$333acAgqsQw'..0AC
14GGGGaz!////aeQU#**,
10@:aCS0SV`abVf0fistuiy0yyyyacACiqs+224Z=M9MPQPST^_`TdPd9d01Jq4D0D:EGHs:VWK[G[:\ ] ] ]a1fqQx'QTAC8??Aac*UV[\X\J]F]`a`cdnorstouwxdy`yFy89JqAvqRSt<T8TGUWXWZ[efgklhlflnqrsnt[uWuGv w w waA%%':aA+>>>>aQ '')ZA-BBBBadCF1H%,,.*SVAXq!t2LLLLa--/1444    c                  &   [        SSS5      n [        SSS5      n[        SS5      n[        SS5      n[        S5      u  pEpg[        U5      R	                  [
        5      S	:X  d   e[        U 5      R	                  [
        5      [        U S-  5      [        U 5      S-  -
  :X  d   e[        XS
9R	                  [
        5      [        U S-  U5      [        X5      S-  -
  :X  d   e[        X5      [        U S-  5      [        U 5      S-  -
  :w  d   e[        X-   5      R	                  [
        5      [        X-   S-  5      [        X-   5      S-  -
  :X  d   e[        X-  5      R	                  [
        5      [        U S-  US-  -  5      [        X-  5      S-  -
  :X  d   e[        X`5      R	                  [
        5      U* [        U 5      -  [        X`-  5      -   :X  d   e[        X5      R	                  [
        5      [        X-  5      [        U 5      [        U5      -  -
  :X  d   e[        XUS
9R	                  [
        5      [        X-  U5      [        X5      [        X5      -  -
  :X  d   e[        S5      u  pdn[        [        X5      5      n[        [        X'5      5      n	[        U 5      R	                  [        5      [        XH-  U[        * [        45      :X  d   e[        U5      R	                  [        5      [        Xy-  US	[        45      :X  d   e[        U 5      R	                  [        5      [        US-  U-  U[        * [        45      [        XH-  U[        * [        45      S-  -
  :X  d   e[        U5      R	                  [        5      [        US-  U	-  US	[        45      [        Xy-  US	[        45      S-  -
  :X  d   e[        X`5      R	                  [        5      U* [        U[        [        X5      5      -  U[        * [        45      -  [        Xd-  [        [        X5      5      -  U[        * [        45      -   :X  d   e[        U 5      R	                  [        5      [        U 5      R	                  [        5      :X  d   e[        U 5      R	                  [        5      [        U 5      R	                  [        5      :X  d   e[        X`5      R	                  [        5      S	:X  d   e[        X`5      R	                  [        5      S	:X  d   e[        XS
9R	                  [        5      [        US-  [        [        X5      U5      -  U[        * [        45      [        U[        [        X5      U5      -  U[        * [        45      S-  -
  :X  d   eg )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&JA!A;{+q000A;{+{16/B[QR^WXEX/XXXXA#++K8KQPQ<RU`abUfjkUk<kkkkA>[A.Q1BBBBBAE?"";/;!|3L{[\[`OaefOf3ffffAE?"";/;qAvQ3OR]^_^cRdhiRi3iiiia##K0QB{1~4ETUTWHX4XXXXa##K0K4D{ST~VabcVdGd4ddddaa(00=QUTUAVYdefYjmxyzm~Y~A~~~~i GA!	RX	B	RX	Bq>!!+.(14!bS"2NNNNq>!!+.#adQ2J2GGGGA;{+x1R!bS"/NQYZ[Z^abegdgik`lQmopQp/ppppA;{+s1a47Q2J/G#adUVXY[]T^J_abJb/bbbba##K02hqRX..RC==[Y[\]YaMbIbefikhkmodp@qqr r r A;s#x{':':8'DDDDq>!!#&+a.*@*@*JJJJa##C(A---a##H-222A#++K8HQT+VXYZV^`aJbEbefikhkmodp<q<DQ{SUVWS[]^G_E_bcfhehjlam<npq<q=r r r rr5   c            
         [        SSS9n [        SSS9n[        S5      n[        SX5      n[        USS	5      nUR                  [        5      [	        US	-
  S-  5      :X  d   eUR                  [
        5      [        US	-
  S-  [        [        X25      5      -  U[        * [        45      :X  d   e[        S
5      n[        [        S	5      US	-
  S-  -  [        UU -
  S	-  * S	US	-  -  -  5      -  S	[        [        5      -  U-  -  U[        * [        45      nUR                  [        5      R                  U5      (       d   eUR                  5       U S-  SU S-  -  -
  SU S	-  -  US	-  -  -   SU S	-  -  -   SU -  US	-  -  -
  SU -  -
  SUS-  -  -   SUS	-  -  -   S-   :X  d   e[        S	S5      nUR                  5       S:X  d   eg )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      s   	D	!BGd+EAsBAq!QA99[![!a%!%<<<<99[!Xq1uqjRX9N.N%&RL&2 2 2 2c
AT!Wa!eaZ'a/. -%(
-$ )% %&'Rj&68:;bS"GE99X''....668A"a%!BE'%(*::2q5b5>*,.rE245eQhJ?ACE1HMOQR S S Sq!A668tr5   c                     [        SSS9n [        SSS9n[        S5      n[        SX5      n[        US5      nUR                  [        5      [	        U[	        U5      -
  S-  5      :X  d   eUR                  [
        5      [        U[        U[        S5      -  U[        * [        45      -
  S-  [        [        X25      5      -  U[        * [        45      :X  d   e[        S	5      n[        [        S
5      U[        [        S
5      U-  [        UU -
  S
-  * S
US
-  -  -  5      -  S
[        [        5      -  U-  -  U[        * [        45      -
  S-  -  [        UU -
  S
-  * S
US
-  -  -  5      -  S
[        [        5      -  U-  -  U[        * [        45      nUR                  [        5      R                  U5      (       d   eUR                  5       R                  5       SUS-  -  :X  d   e[!        SS5      nUR                  5       S:X  d   eg )Nr<   Tr#   r=   r>   r"   r   rB   r@   r      r&   i5  )r	   r   r   r+   r   r   r   r   r   r   r   r
   r   rF   r,   simplifyr   )r<   r=   r"   r   CMr@   rH   s          r3   test_symbolic_CentralMomentrN      s&   	D	!BGd+EAsBA	q!	B::k"k1{1~3E2I&JJJJ::k"hHQ{4?P=PRL5" 1"$%0&&1"Q(&;0<>?"b\'K K K Kc
AT!Wa(4719S1
<; :%(
: 6 ,Rj.,023bS"#? ?ABC CCF
JI H%(
H DRj.023bS"?E ::h((////7792eQh;...	1B779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\      sP    )  ' $ . 6 9 8 . %J J J 3P5f&rR&r5   