Is the order of random variables important in the chain rule? I mean, is this true: $P(A,B,C) = P(A)\times P(B|A)\times P(C|A,B) = P(C)\times P(B|C)\times P(A|B,C) = P(C,B,A)$? If it is, what is the meaning of such order? Thank you.
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$P[A \cap B \cap C] = P[(A \cap B) \cap C] = P[(A \cap B)|C]P(C) = P[C|A \cap B]P[A \cap B]$. Then you can rewrite $P(A \cap B) = P(A|B)P(B) = P(B|A)P(A)$.
These are all useful. Suppose you want to find $P(A \cap B)$. Well $P(A \cap B) = P(A|B)P(B) = P(B|A)P(A)$. But suppose you only know $P(B|A)$. Then $P(B|A)P(A)$ is more useful.
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1So the ordering is unimportant, right? I just want to confirm this! – Martin08 Jan 26 '11 at 23:30
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1@Martino8: Yes it is unimportant. – NebulousReveal Jan 26 '11 at 23:40
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@NebulousReveal Is it also unimportant for random variables? I mean does it matter if we write it as $P(X_1, X_2, X_3)$ or $P(X_3, X_2, X_1)$? – Antonios Sarikas Mar 11 '22 at 18:22