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Let $X,Y,Z$ denote random variables and we define the conditional random variable $W:=Y|X$. Is it true, that $$W|Z=(Y|X)|Z = Y|X,Z,$$ i.e. $$P(W|Z)=P((Y|X)|Z)=P(Y|X,Z)\quad ?$$ Can you provide me maybe a valid proof? Thanks!

Henry
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anjo1659
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  • Thanks @Henry! Random variables are usually associated with their distributions. Therefore, I did not see any need to clarify this more. But I can follow your suggestion. – anjo1659 Apr 27 '25 at 10:07
  • See https://math.stackexchange.com/questions/364620/double-conditional-probability – Henry Apr 27 '25 at 10:19

1 Answers1

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Conditional probability is defined as

$$P(A|B) = \frac{P(A \cap B)}{P(B)}$$

which is simply the intersection of events, so in your case

$$P((Y|X)|Z) = P(Y|X,Z) = \frac{P(Y \cap X \cap Z)}{P(X \cap Z)}$$

k1r1t0
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