I would like to prove that E(X |X< Y) <= EX where X and Y are independent random variables. It's easy to prove the inequality when Y is a constant, but what about when Y is only independent from X? We assume that X,Y have means and P(X< Y)>0.
2026-03-29 19:17:39.1774811859
Proving E(X|X<Y) that is less than or equal to EX where X and Y are independent random variables
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If you know that $EXI_{X<t} \leq (EX) P(X<t)$ for all $t$ you can just integrate this w.r.t the distribution of $Y$. Fubini's Theorem gives you $EXI_{X<Y} \leq (EX) P(X<Y)$.