Symmetric distributions in $\mathbb{R}$.

Question

Let $X$ be a real random variable central symmetric with respect to $c\in\mathbb{R}$, that is, $X - c$ is equal in distribution to $c - X$, that is,

$$ \mathbb{P}(X - c\leq t) = \mathbb{P}(c - X\leq t), $$ for all $t\in\mathbb{R}$.

It is clear that $c$ is unique and $c$ is a median of $X$.

In https://en.wikipedia.org/wiki/Symmetric_probability_distribution they say that $c$ also coincides with $\text{E}[X]$ (if it exists). Why?

I've tried to use the definition of the expectation using cummulative distribution function, but i don't see clear the proof.

Thanks in advance.

Answer 1

If $X-c$ and $c-X$ have the same distribution, then $$\mathbb E[X]-c=\mathbb E[X-c]=\mathbb E[c-X]=c-\mathbb E[X]\implies\mathbb E[X]=c.$$