TITLE: $E(Z_1^2|Z_1+Z_2=s)$ where $Z_1,Z_2$ are iid. normal distribution
QUESTION [0 upvotes]: I'm looking for the fastest way to compute the conditional expectation. I'm mostly stuck with $E(Z_1Z_2|Z_1+Z_2=s)$. I'm not sure if $E(Z_1Z_2|Z_1+Z_2=s)=E(Z_1|Z_1+Z_2=s)E(Z_2|Z_1+Z_2=s)$. Any ideas except for brute force calculus would be helpful.

REPLY [0 votes]: The event $Z_1+Z_2=s$ is null, so you are conditioning on a null event, which by convention is defined to be $0$.
If you want $\mathbb{E}[Z_1^2\mid Z_1+Z_2]$ instead, then do not write it as $\mathbb{E}[Z_1^2\mid Z_1+Z_2=s]$, and use the rotation as in grand_chat's solution.
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