Suppose $X,Y,Z$ are non negative random variables. Is it possible that $X\le Y$ almost surely, but $V(X\mid \sigma(Z))+E(X\mid \sigma(Z))>V(Y\mid \sigma(Z))+E(Y\mid \sigma(Z))$ almost surely? I think yes, but I could not find an example. Could someone provide one?
2026-04-07 01:42:16.1775526136
$X\le Y$ but $V(X\mid Y)>V(Y\mid Y)$
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$Z=1$, $X$ is Bernoulli with probability $p \in (0,1)$ and $Y=1$.