This is not a homework problem but is rather something I'm curious about. I apologize if the answer is very simple, but I am having trouble coming up with an absolute and strict proof. * X is a discrete random variable that is symmetrically distributed about 0. Hence, E(X) = 0 * Why is E(X^3) = 0 but E(X^2) is not necessarily zero? * My intuition is that this is because X^3 and X are both odd functions. How would I write this out in equation form? Do I have to apply the law of total expectation (aka. law of iterated expectations)? Any thoughts and suggestions would be appreciated. Thank you!