Questions on the law of total multivariate probability












0














I am wondering whether these formulations are established:



$P(Y|X,A)=int_Z P(Y|Z,A)P(Z|X,A) dZ$



$P(Y|X,A)=int_Z P(Y|Z)P(Z|X,A) dZ$



The $Y$, $A$, $Z$ are all random variables.



And if so, how to prove and comprehend them?



thanks so much!










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  • 1




    What exactly is meant with the notations like $p(Ymid X,A)$ and $P(Ymid Z,A)$? Also: is there any difference between e.g. $p(Ymid X,A)$ and $P(Ymid X,A)$? In the integrals on RHS variable $Z$ seems to be overloaded with meanings.
    – drhab
    Dec 26 at 11:13












  • The $p(Y|X,A)$ and $P(Y|X,A)$ are the same. I am sorry about making this typo.
    – YRS
    Dec 26 at 12:17










  • BTW, could you explain the meaning of variable $Z$ overloaded?
    – YRS
    Dec 26 at 12:19










  • I see $Z$ as part of integral sign ($int_Z$, why?) and also as intergrating variable ($dZ$). Do the notations stand for conditional PDF'S maybe? Is $Z $ a random variable or a normal variable? Questions like this cross my mind.
    – drhab
    Dec 26 at 12:24












  • How to integrate over events (I mean $Z $)? For that it must be a variable.
    – drhab
    Dec 26 at 12:32
















0














I am wondering whether these formulations are established:



$P(Y|X,A)=int_Z P(Y|Z,A)P(Z|X,A) dZ$



$P(Y|X,A)=int_Z P(Y|Z)P(Z|X,A) dZ$



The $Y$, $A$, $Z$ are all random variables.



And if so, how to prove and comprehend them?



thanks so much!










share|cite|improve this question









New contributor




YRS is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
















  • 1




    What exactly is meant with the notations like $p(Ymid X,A)$ and $P(Ymid Z,A)$? Also: is there any difference between e.g. $p(Ymid X,A)$ and $P(Ymid X,A)$? In the integrals on RHS variable $Z$ seems to be overloaded with meanings.
    – drhab
    Dec 26 at 11:13












  • The $p(Y|X,A)$ and $P(Y|X,A)$ are the same. I am sorry about making this typo.
    – YRS
    Dec 26 at 12:17










  • BTW, could you explain the meaning of variable $Z$ overloaded?
    – YRS
    Dec 26 at 12:19










  • I see $Z$ as part of integral sign ($int_Z$, why?) and also as intergrating variable ($dZ$). Do the notations stand for conditional PDF'S maybe? Is $Z $ a random variable or a normal variable? Questions like this cross my mind.
    – drhab
    Dec 26 at 12:24












  • How to integrate over events (I mean $Z $)? For that it must be a variable.
    – drhab
    Dec 26 at 12:32














0












0








0







I am wondering whether these formulations are established:



$P(Y|X,A)=int_Z P(Y|Z,A)P(Z|X,A) dZ$



$P(Y|X,A)=int_Z P(Y|Z)P(Z|X,A) dZ$



The $Y$, $A$, $Z$ are all random variables.



And if so, how to prove and comprehend them?



thanks so much!










share|cite|improve this question









New contributor




YRS is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.











I am wondering whether these formulations are established:



$P(Y|X,A)=int_Z P(Y|Z,A)P(Z|X,A) dZ$



$P(Y|X,A)=int_Z P(Y|Z)P(Z|X,A) dZ$



The $Y$, $A$, $Z$ are all random variables.



And if so, how to prove and comprehend them?



thanks so much!







probability






share|cite|improve this question









New contributor




YRS is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.











share|cite|improve this question









New contributor




YRS is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.









share|cite|improve this question




share|cite|improve this question








edited Dec 26 at 15:01





















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asked Dec 26 at 10:50









YRS

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42




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YRS is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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New contributor





YRS is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.






YRS is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.








  • 1




    What exactly is meant with the notations like $p(Ymid X,A)$ and $P(Ymid Z,A)$? Also: is there any difference between e.g. $p(Ymid X,A)$ and $P(Ymid X,A)$? In the integrals on RHS variable $Z$ seems to be overloaded with meanings.
    – drhab
    Dec 26 at 11:13












  • The $p(Y|X,A)$ and $P(Y|X,A)$ are the same. I am sorry about making this typo.
    – YRS
    Dec 26 at 12:17










  • BTW, could you explain the meaning of variable $Z$ overloaded?
    – YRS
    Dec 26 at 12:19










  • I see $Z$ as part of integral sign ($int_Z$, why?) and also as intergrating variable ($dZ$). Do the notations stand for conditional PDF'S maybe? Is $Z $ a random variable or a normal variable? Questions like this cross my mind.
    – drhab
    Dec 26 at 12:24












  • How to integrate over events (I mean $Z $)? For that it must be a variable.
    – drhab
    Dec 26 at 12:32














  • 1




    What exactly is meant with the notations like $p(Ymid X,A)$ and $P(Ymid Z,A)$? Also: is there any difference between e.g. $p(Ymid X,A)$ and $P(Ymid X,A)$? In the integrals on RHS variable $Z$ seems to be overloaded with meanings.
    – drhab
    Dec 26 at 11:13












  • The $p(Y|X,A)$ and $P(Y|X,A)$ are the same. I am sorry about making this typo.
    – YRS
    Dec 26 at 12:17










  • BTW, could you explain the meaning of variable $Z$ overloaded?
    – YRS
    Dec 26 at 12:19










  • I see $Z$ as part of integral sign ($int_Z$, why?) and also as intergrating variable ($dZ$). Do the notations stand for conditional PDF'S maybe? Is $Z $ a random variable or a normal variable? Questions like this cross my mind.
    – drhab
    Dec 26 at 12:24












  • How to integrate over events (I mean $Z $)? For that it must be a variable.
    – drhab
    Dec 26 at 12:32








1




1




What exactly is meant with the notations like $p(Ymid X,A)$ and $P(Ymid Z,A)$? Also: is there any difference between e.g. $p(Ymid X,A)$ and $P(Ymid X,A)$? In the integrals on RHS variable $Z$ seems to be overloaded with meanings.
– drhab
Dec 26 at 11:13






What exactly is meant with the notations like $p(Ymid X,A)$ and $P(Ymid Z,A)$? Also: is there any difference between e.g. $p(Ymid X,A)$ and $P(Ymid X,A)$? In the integrals on RHS variable $Z$ seems to be overloaded with meanings.
– drhab
Dec 26 at 11:13














The $p(Y|X,A)$ and $P(Y|X,A)$ are the same. I am sorry about making this typo.
– YRS
Dec 26 at 12:17




The $p(Y|X,A)$ and $P(Y|X,A)$ are the same. I am sorry about making this typo.
– YRS
Dec 26 at 12:17












BTW, could you explain the meaning of variable $Z$ overloaded?
– YRS
Dec 26 at 12:19




BTW, could you explain the meaning of variable $Z$ overloaded?
– YRS
Dec 26 at 12:19












I see $Z$ as part of integral sign ($int_Z$, why?) and also as intergrating variable ($dZ$). Do the notations stand for conditional PDF'S maybe? Is $Z $ a random variable or a normal variable? Questions like this cross my mind.
– drhab
Dec 26 at 12:24






I see $Z$ as part of integral sign ($int_Z$, why?) and also as intergrating variable ($dZ$). Do the notations stand for conditional PDF'S maybe? Is $Z $ a random variable or a normal variable? Questions like this cross my mind.
– drhab
Dec 26 at 12:24














How to integrate over events (I mean $Z $)? For that it must be a variable.
– drhab
Dec 26 at 12:32




How to integrate over events (I mean $Z $)? For that it must be a variable.
– drhab
Dec 26 at 12:32















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