The variance can be written as Var[X]=E[X^2]-(E[X])^2. Use this form to prove that the Var[X] is always non-negative, i.e., show that E[X^2]>=(E[X])^2. Use Jensen's Inequality. Any sugestions? I just tried to prove that a function g(t) is continuous and twice differentiable, such that g''(t) > 0 which must imply it is convex. Then, I am stuck with the proof.