TITLE: Prove that random variables satisfy the inequality $E(XY)^2 \le E(X^2)E(Y^2)$?
QUESTION [13 upvotes]: Given Random variables $X$ and $Y$ is it true always that;
$$E(XY)^2 \le  E(X^2)E(Y^2)$$
Is it easy to prove?

REPLY [3 votes]: The expectation of a product of random variables is an inner product, to which you can apply the Cauchy-Schwarz inequality and obtain exactly that inequality. Hence the answer is yes.
See http://en.wikipedia.org/wiki/Cauchy%E2%80%93Schwarz_inequality#Probability_theory