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(Solved): Suppose \( P(X, Y)=f(X) g(Y) \) for some functions \( f \) on \( X \) and \( ...



Suppose \( P(X, Y)=f(X) g(Y) \) for some functions \( f \) on \( X \) and \( g \) and \( Y \). Prove that \( X \perp Y \).???????

Suppose \( P(X, Y)=f(X) g(Y) \) for some functions \( f \) on \( X \) and \( g \) and \( Y \). Prove that \( X \perp Y \).


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f(x)g(Y) = P(A, Y)+P(A', Y)-P(A', Y) = P(A, Y) Explanation: P(X, Y) = P(Y|X)P(X). Let A be any subset of X. You have P(A, Y) = P(Y|A)P(A) = P(Y|A)P(X|A)P
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