Showing 1/E(W) <= E(1/W)

Question

How do I show that $\displaystyle \frac{1}{E(W)} \leq E\left(\frac{1}{W}\right)$ for a positive random variable W?

I may be intended to use the Cauchy-Schwarz Inequality, $[E(XY)]^2 \leq E(X^2)E(Y^2)$.

Answer 1

Take $X=1/\sqrt{W}$ and $Y = \sqrt{W}$ and you get what you want.

Answer 2

$E[1]^2\leq E[(\sqrt{W})^2]E[(1/\sqrt{W})^2]$ so $1/E[W]\leq E[1/W]$.