Conditional expectation for independent gaussians

Question

This is an interview question I found online.

Let $X$ and $Y$ be independent standard Gaussians. What is $E[X|XY]$?

I don't know how the expectation function distributes over conditionals. Any tips?

Answer 1

Here is a hint: Note that $X$ and $Y$ are symmetric distributions. Suppose you are told that the product of $X$ and $Y$ is, say, $1$. Does that give you preferential information about the sign of $X$; that is to say, is the value of $XY$ informative about whether $X$ is positive or negative?

Do you have reason to believe the conditional distribution of $X \mid XY$ would be asymmetric?