Suppose $X,Y,Z$ are three independent standard normal variables, what are the conditional expected value and variance of $X$, i.e., $E[X|X+Y+Z=3]$? $\text{Var}(X|X+Y+Z=3)$?
2026-04-11 12:55:28.1775912128
Conditional expected value $E[X|X+Y+Z=3]$
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Hint: $~\mathsf E\big(X+Y+Z~\big\vert~ X+Y+Z=3\big)~=~3$
Follow with Linearity of Expectation and an argument for symmetry.