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Let $A,B,C$ be any (continuously distributed, if you need) random variables such that $A$ is independent of $(B,C)$ (B and C may be correlated). I want to show that (or find sufficient conditions for)

$$ E(A \mid A\leq B) \geq E(A \mid A\leq \min(B,C)). $$

I expected this should be trivially true, but couldn't prove it despite my best effort for weeks to the point that I am not even sure if it's true anymore...

Can anyone provide proof or a counter-example, please? When proving, you are welcome to make any regularity assumptions such as random variables being continuously distributed and having finite moments, etc., if necessary.

lee
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  • I'm pretty sure you just need $A\geq 0$ Because then you just have a monotonicity thing going, where the restricted sample space of $A\leq \min (B,C)$ is a subset of $A\leq B$, and integrating a nonnegative function over a bigger space can only make it bigger – Alan Aug 15 '21 at 12:09
  • @Alan You are forgetting that $E(A|E)=\frac {E(AI_E)} {P(E)}$ if $E$ is an event and $A$ is a r.v. Your argument fails even when $A \geq 0$. – Kavi Rama Murthy Aug 15 '21 at 12:15
  • @KaviRamaMurthy Check, thus the pretty sure. Probability is one of my weakest areas! Thanks – Alan Aug 15 '21 at 15:34

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