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What's the maximum expectation of a conditional variance, $E[\operatorname{Var}(X+Z_1 \mid X+Z_2)]$?
Let $X,Z_1,Z_2$ be 3 mutually independent RV's, with $Z_1, Z_2$ assuming $N(0,1)$ distribution. $X$ is constrained to have unit 2nd moment, i.e. $E[X^2] =1$, but may take arbitrary distribution. The ...
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