Inequality between Probability and Expectation

Question

I have to prove an inequality between probability and expectation and I wanted to ask for help on it. Here is the problem:

Assume that $Y \ge 0$ and $E Y^2 < \infty$. I need to prove that:

$$\Pr(Y > 0) \ge \frac{(EY)^2}{(EY^2)}$$

Thanks for your help and hints in advance.

Answer 1

Hint: Apply Cauchy-Schwarz inequality to $Y=Y\cdot\mathbf 1_{Y\gt0}$.