#StackBounty: #probability #self-study #expected-value #mean-absolute-deviation an upper bound of mean absolute difference?

Bounty: 50

Let $X$ be an integrable random variable with CDF $F$ and inverse CDF $F^*$. $Y$ is iid with $X$. Prove $$E|X-Y| leq frac{2}{sqrt{3}}sigma,$$ where $sigma=sqrt{Var(X)} = sqrt{E[(X-mu)^2]}$.

I am looking for some hint for this proof.

What I’ve got is $E|X-Y|=2int_{0}^{1}(2u-1)F^*(u)du$. But I am not sure if this is correct direction.

I also noticed that $frac{2}{sqrt{3}}$ may be related to the variance of the uniform distribution.

Get this bounty!!!

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