Recall that the Cauchy–Schwarz inequality stats that for any two random variable \(X\) and \(Y\) one has that

\[ \mathbf E |XY| \leq \sqrt{\mathbf E [X^2]}\,\sqrt{ \mathbf E [Y^2]}\]

- Use it to show that

\[ \mathbf E |X| \leq \sqrt{\mathbf E [X^2]}\]