symmetric random variable

Let $(\mathrm{\Omega },ℱ,P)$ be a probability space and $X$ a real random variable defined on $\mathrm{\Omega }$. $X$ is said to be symmetric if $-X$ has the same distribution function as $X$. A distribution function $F:ℝ\to [0,1]$ 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 (). 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 $x\in ℝ$, 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 $\mathrm{\Delta }X$ of two independent random variables, identically distributed as $X$ is symmetric.