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I know that independent and conditional independent don't imply each other. But what if given more condition that $Z$ is independent from $X$ and $Z$ is independent from $Y$?

So the problem is:

A: Random Variables $X$ and $Y$ are independent.

B: $X$ and $Y$ are independent given condition $Z$. $Z$ is independent from $X$ and also $Z$ is independent from $Y$.

Can B $\implies$ A be true? (Given B, can we conclude that A is true?)

Thanks for helping me prove or disprove it. I tried it by myself but only found that A is true if adding "$Z$ is also independent from $X,Y$" condition to B.

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1 Answer

Consider the probability space consisting of six equally likely outcomes $a,b,c,d,e,f$. Let $Z$ be the event $\{a,b,c,d\}$, and let $X$ and $Y$ be the random variables with values given by the following table:

$$\matrix{ \text{outcome} & X & Y\cr a & 1 & 1\cr b & 0 & 1\cr c & 1 & 0\cr d & 0 & 0\cr e & 1 & 1\cr f & 0 & 0\cr}$$

Given $Z$, $X$ and $Y$ are independent: e.g. $P(X=0,Y=0 | Z) = 1/4 = P(X=0|Z) P(Y=0|Z) = (1/2) (1/2)$.

$X$ and $Z$ are independent: $P(X=0,Z) = 1/3 = P(X=0) P(Z) = (1/2)(2/3)$, and similarly $Y$ and $Z$ are independent.

However, $X$ and $Y$ are not independent: $P(X=0,Y=0) = 1/3 \ne P(X=0) P(Y=0) = 1/4$.

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