Expected Value Inequality - bounded by sums of probabilitys

Question

I have to show the following inequality: $$ \sum_{k=0}^{\infty} P[Y>k] \leq E[Y] \leq \sum_{k=1}^{\infty} P[Y>k]$$ under the condition that Y is a non negative continuous random Variable.

My first idea was to rewrite the inequality using integrals, but then i end up having to integrate over a sum, which is nothing im particulary excited about. Any Ideas on how i can do this?

Answer 1

Hint: If $Y$ is nonnegative, then $$E[Y] = \int Y \, dP = \int \int_0^\infty I\{x < Y\} \, dx \, dP = \int_0^\infty \int I\{x < Y\} \, dP \, dx = \int_0^\infty P(Y > x) \, dx$$

Now apply the standard comparison between integrals and sums.