Does $| E[X*1\{X<\mu\}] | = | E[X*1\{X>\mu\}] |$?

Question

Does mean of a random variable $\mu=E[X]$ devide conditional mean in half, in a sense that

$$ | E]X*1\{X<\mu\}] | = | E[X*1\{X>\mu\}] |$$

or is this only true for symmetric distributions?

Answer 1

Not true in general. Let $X$ take values $1$ and $2$ with ptobabilites $\frac 1 3$ and $\frac 2 3$. Then EX=$\frac 5 3$, LHS is $\frac 1 3$ and RHS is $\frac 4 3$

Answer 2

This is not even true for symmetric distributions. Suppose that $P(X=-1)=P(X=1)=1/2$. Then $$ \operatorname E[X1\{X<0\}]=-\frac12 $$ but $$ \operatorname E[X1\{X>0\}]=\frac12. $$