By Definition of Expectation of Random Variable:
$$ E(X)= \int_{-\infty}^{\infty}\,x\,f_X(x)\,dx $$
Now if the pdf $f_X(x)$ is Even we know that $E(X)=0$ (Ofcourse if integral Converges, i.e, Lets exclude cases like Cauchy Random Variable)
Is the Converse True, i.e., is there a Random Variable $X$ whose pdf is Neither Even-Nor Odd, such that $E(X)=0$.
Best Answer
It is possible to make up as many examples as you wish. Let $Y$ be almost any random variable with mean $\mu$, and let $X=Y-\mu$. All we need to do is to avoid symmetry about $\mu$, so for example let $Y$ have exponential distribution, or density $2y$ in $[0,1]$ and $0$ elsewhere.
For a discrete example, let $Y$ have binomial distribution with $p\ne 0$, $1$, or $1/2$.