I was told the following fact
For a non-negative random variable $X$, we have
$$
\mathbb{E}(X) < \infty \ \ \ \text{if and only if} \ \ \ \sum_{n=0}^\infty \mathbb{P}(X \geq n) < \infty
$$
The only if part is clear to me, as
$$
\mathbb{E}(X) = \int_0^\infty \mathbb{P}(X \geq t) dt \geq \sum_{n=0}^\infty \mathbb{P}(X \geq n)
$$
Yet, I have no clue how to approach the other side. Any hints are welcome, thanks!
Best Answer
Integrating $P(X \ge \lceil t \rceil) \le P(X \ge t) \le P(X \ge \lfloor t \rfloor)$ yields $$ \sum_{n = 1}^\infty P(X \ge n) \le \int_0^\infty P(X \ge t) \, dt \le \sum_{n = 0}^\infty P(X \ge n).$$