If X is a continuous random variable symmetric about some point a, how do I show that E(X) = a? Thanks.
Break the integral into two parts.
$\displaystyle E(X)=\int_{-\infty}^axf(x)dx+\int_a^{\infty}xf(x)dx$
Let $\displaystyle x=a-u$ in the first integral and $\displaystyle x=a+u$ in the second.
Note that $\displaystyle f(a-u)=f(a+u)$