# Conditional Covariance Problem

Suppose we have independent (not necessarily identical) normally distributed random variables X, Y. If we're given that, upon sampling each variable, X is some multiple a of Y (i.e. x = ay), what is the covariance of X and Y?

I know we can solve this by finding the conditional joint probability distribution via Bayes' Law and calculating the conditional covariance by integrating on this joint pdf, but I'm wondering if there's a cleaner way to do this.

• Welcome to Stats.SE. Can you please explain what do you mean by "independent (not necessarily identical) [...]" with "$X$ is some multiple $a$ of $Y$ (i.e. $x = ay$)"? – Ertxiem - reinstate Monica May 24 at 23:23
• "independent normally distributed random variables X, Y " and "X is some multiple a of Y" cannot be true simultaneously. So you need to learn more about some basic concepts. – user158565 May 25 at 0:19