# symmetric random variable

## Primary tabs

Defines:
symmetric distribution function
Type of Math Object:
Definition
Major Section:
Reference
Groups audience:

## Mathematics Subject Classification

The remark according to which if a random variable $X$ is symmetric, then $E[X]$ exists, does not seem to be correct. For example, if a random variable has a Cauchy–Lorentz distribution, i.e., its probability density function is
 $f(x)=\frac{1}{\pi\gamma[1+(\frac{x-x_{0}}{\gamma})^{2}]},$
with location parameter $x_{0}=0$ and scale parameter $\gamma>0$, then it is symmetric, but its mean is undefined.
I think that the correct statement is that if $X$ is symmetric *and* $E[X]$ exists, then $E[X]=0$.