Conditional Independence problem.

Question

Given,
$X = P + Q$,

$Y = P + R$ such that $P$, $Q$, $R$ are independent.

Now, how can I show that $X$ and $Y$ are conditionally independent given $P$?

Thanks in advance.

Answer 1

Well, $\mathsf P(X\,{\leq}\, x, Y\,{\leq}\,y\mid P\,{=}\,z)=\mathsf P(Q\,{\leq}\,x-z,R\,{\leq}\,y-z\mid P\,{=}\,z)$, so ... start there.