About Conditional Independent 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.
A: 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: $P(X=0,Y=0) = 1/3 \ne P(X=0) P(Y=0) = 1/4$.
