Expectation of the absolute value of $X$ if $E[X]=0$

Question

My question is rather short:

what is the expectation of the absolute value of $X$ (or, $E[|X|]$) if $E[X]=0$?

Answer 1

As you may have found from the comments, this doesn't give you much information about the expectation of $\lvert X \rvert$.

Indeed, let $X$ be a random variable and $a$ be some nonnegative constant. Then we can let $X$ have the discrete distribution such that $P(X = -a) = P(X = a) = \frac 12$, so clearly $E[X] = 0$, but also $P(\lvert X \rvert = a) = 1$ so $E[\lvert X \rvert] = a$ (this can be adjusted a little depending on whether you're working with discrete/continuous random variables etc, but you get the idea).