Expected difference from mean: is it always zero? The quantity
$$
\mathbf{E}(x-\mu)=\int (x-\mu)P(x) dx
$$
is equal to zero for symmetric probability density functions. What about the others?

If $\mu = \mathbb{E}[X]$, then
$$
\mathbb{E}[X - \mu] = \mathbb{E}[X] - \mathbb{E}[\mu] = \mathbb{E}[X] - \mu = \mu - \mu = 0
$$
regardless of the skewness of $X$
