What can I infer about a correlation based on another correlation that shares one variable?

I have three vectors, $X$, $Y$, and $Z$. Each element of $X$ is independently normally distributed $X_isim N(0,sigma^2_X)$.

The elements of $Y$ and $Z$ are jointly normally distributed:

$left[ begin{array}{c}Y_i\ Z_iend{array}right]sim Nleft(left[begin{array}{c}0\ 0end{array}right],left[begin{array}{cc}sigma^2_Y & rho \ rho & sigma^2_Zend{array}right]right)$

So, naturally, $E(X’Y)=E(X’Z)=0$.

However, in finite samples, $X’Y$ doesn’t have to be zero.

What can I say about $E(X’Z)$ conditional on my sample observation of $X’Y$? In other words, can I say anything about $E(X’Z|X’Y)$?

I feel like this is probably not a super difficult problem but I’ve come at it a few different ways and not really been able to make much headway (aside from a simulation, which gives me an answer but not why, or how conditional it is on the parameters I set), which makes me think I’m missing something obvious.

While in the middle of writing this up it occurred to me that I could probably calculate a distribution of $X’Z$ conditional on $X’Y$ from the Wishart distribution. But I, uh, hope it’s simpler than that.

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