If X,Y are two random variables such that for every integrable increasing function $f$ $$E(f(X))\leq E (f(Y))$$ can I say that $X\leq Y$ almost surely?
2026-04-21 16:25:51.1776788751
Expected value and function of random variables
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Consider two independent random variables with 0-1 distribution: $X$ with $P(X=0)=1-s$ and $P(X=1)=s$, $Y$ with $P(Y=0)=1-t$ and $P(Y=1)=t$. A condition in question translates to $(t-s)\bigl(f(1)-f(0)\bigr)\ge 0$ which is equivalent to $s\le t$. What about $P(X\le Y$)?