conditional expectation of exponentials of independent random variables

Question

For two independent identically distributed random variables $X$ and $Y$, we have $$E[e^{iX-iY}]=E[e^{iX}]E[e^{-iY}]=|E[e^{iX}]|^2.$$ I am wondering whether the above property still holds for conditional expectation, that is, if there is another random variable $Z$ which is independent of $X$ and $Y$, is the following true: $$E[e^{iZX-iZY}|Z]=|E[e^{iZX}|Z]|^2.$$ If true, how to prove it? I know that $E[XZ|Z]=ZE[X|Z]=ZE[X]$, but how to compute $E[e^{iZX}|Z]$? Thanks for helping!

Answer 1

If $Z$ is independent of $X$ and $Y$ then $X$ and $Y$ will still be conditionally independent given $Z,$ and the same calculation you did before goes through.