What is $E[E[X|Y]|X]$? Is it just $X$? Any hint?
2026-03-30 12:56:03.1774875363
$E[E[X|Y]|X]$ please help
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This will depend on how $X$ and $Y$ are related.
Considering the extremes, if $X=Y$ (or more generally, $\sigma(X) = \sigma(Y)$), then $$E\Big[E[X|Y]\Big|X\Big] = E\Big[E[X|X]\Big|X\Big] = E\Big[X\Big|X\Big] = X$$ while if $X,Y$ are independent $$E\Big[E[X|Y]\Big|X\Big] = E\Big[E[X]\Big|X\Big] = E\Big[X\Big]$$