How to prove or disprove the following statement? How to prove or disprove the following statement:
"If random variables X and Y are independent, AND if random variables Y and Z are independent, then X and Z are independent."
I tried to use Bayes Rule to prove it but it doesn't work. I think it might actually be wrong... but I just cannot think about an example...
Thanks for your help!

All you need is an example of
$P(X)P(Y)=P(X\cap Y)$
$P(Y)P(Z)=P(Y\cap Z)$
$P(X)P(Z)\neq P(X\cap Z)$
Consider six mutually exclusive events {$s_1,s_2,s_3,s_4,s_5,s_6$} each with probability $1\over6$ (for example a dice)
Let
$X=\{s_1,s_2,s_5\}$ (any one of them happening)
$Y=\{s_1,s_2,s_3,s_4\}$
$Z=\{s_1,s_2,s_6\}$
Then
$P(X)P(Y)=P(X\cap Y)={1\over3}$
$P(Y)P(Z)=P(Y\cap Z)={1\over3}$
$P(X)P(Z)={1\over4}\neq P(X\cap Z)={1\over3}$
