Inequality between the expected value of the absolute value of a random variable and the expect value of the variable itself.

Question

Amid one exercise I am solving, it would be extremely helpful to guarantee that the inequality

$$ E[ \, |X| \, ] \leqslant | \, E[X] \, | $$

holds. So, essentially, I would like to know under what conditions (if any) is this inequality true (or, in other words, what conditions do I have to impose to the r.v. $X$ (if any) such that this inequality holds).

Thanks for any help in advance.

Answer 1

$0\leq E(|X|-X)=E(|X|)-E(X)$ and equality implies $|X|-X=0$ with pb 1, or $\Pr(X\geq 0)=1.$ Similarly $0\leq E(|X|+X)=E(|X|)+E(X)$ and equality implies $|X|+X=0$ with pb $1,$ or $\Pr(X\leq 0)=1.$ Now $E(|X|)\leq |E(X)|$ implies $E(|X|)= |E(X)|$ since we have always $E(|X|)\geq |E(X)|.$ Therefore $E(|X|)\leq |E(X)|$ holds if and only if either $\Pr(X\geq 0)=1$ or $\Pr(X\leq 0)=1.$