Can we compute $$ E\big[ E[Y \mid X] \cdot E[Y \mid X] \big] $$ in closed form? Supposing that $X$ takes values in $S$ and $Y$ takes values in $T$ (notation from randomservices.org), I've tried expanding according to the definitions, but it seems it does not simplify, $$ E\big[ E[Y \mid X] \cdot E[Y \mid X] \big] = \int_S \bigg(\int_T y P(Y=y \mid X=x)dy \bigg) \cdot \bigg(\int_T z P(Y=z \mid X=x)dz \bigg) P(X=x)dx $$ Am I missing something?
2026-04-17 18:37:46.1776451066
expectation of squared conditional expectation
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