Let $(\Omega,\mathcal{F},P)$ be a probability space and $X$ a real random variable defined on $\Omega$ $X$ is said to be symmetric if $-X$ has the same distribution function as $X$ A distribution function $F:\mathbb{R}\to [0,1]$ is said to be symmetric if it is the distribution function of a symmetric random variable.

Remark. By definition, if a random variable $X$ is symmetric, then $E[X]$ exists ($<\infty$ . Furthermore, $E[X]=E[-X]=-E[X]$ so that $E[X]=0$ Furthermore, let $F$ be the distribution function of $X$ If $F$ is continuous at $x\in\mathbb{R}$ then $$F(-x)=P(X\le -x)=P(-X\le -x)=P(X\ge x)=1-P(X\le x)=1-F(x),$$ so that $F(x)+F(-x)=1$ This also shows that if $X$ has a density function $f(x)$ then $f(x)=f(-x)$

There are many examples of symmetric random variables, and the most common one being the normal random variables centered at $0$ For any random variable $X$ then the difference $\Delta X$ of two independent random variables, identically distributed as $X$ is symmetric.