Prove that $E(\sqrt{X})\leq\sqrt{EY}$ Suppose that $X > 0$ and $Y > 0$ are random variables such that $E(X/Y) ≤ 1$. Prove that
$E(\sqrt{X})\leq\sqrt{EY}$
Note: Here X/Y is X divided by Y .
Since I don't know whether X and Y are independent so I cannot separate them from E(X/Y). How to solve this problem?

Write $\sqrt{X}$ as $\sqrt{X/Y}\cdot\sqrt{Y}$ and use the Cauchy-Schwarz inequality to obtain the result.
