Finding the expected value of a nonnegative integer valued random variable

Question

If $X$ is a nonnegative integer valued random variable, show that

$$\mathbb{E}(X) = \sum_{i=1}^\infty\mathbb{P}(X\geq i) = \sum_{i=0}^\infty \mathbb{P}(X\geq i). $$

I'm not sure how to do this. I only know the definition that $$\mathbb{E}(X):= \sum_{i=-\infty}^\infty i \mathbb{P}(X=i)\mathbf{1}_{\{X\geq 0\}} = \sum_{i=1}^\infty i \mathbb{P}(X=i). $$

There is a mistake in the statement. The last sum must have $P{X>i}$ instead of $P{X \geq i}$ — Kavi Rama Murthy, Mar 08 '18 at 07:28
Despite being already closed as a duplicated, this comment is to link this post to the current choice of mother post. Please see the (abstract) duplicates — Lee David Chung Lin, Nov 13 '18 at 13:31

score 0 · Answer 1 · answered Mar 08 '18 at 03:28

0

Assert that most surely $X\in \Bbb N^+$ , then use that for all $k\in\Bbb N^+$, by definition of counting: $k=\sum_{j=0}^{k-1} 1$ .

$$\mathsf E(X) {= \sum_{k=1}^\infty k~\mathsf P(X=k) \\ = \sum_{k=1}^\infty\mathsf P(X=k)\sum_{j=0}^{k-1}1\\ \vdots \\ = \sum_{j=0}^\infty \mathsf P(X>j)}$$

answered Mar 08 '18 at 03:28

Graham Kemp

129,094

What if $E[X] = \infty$ ? – BCLC Mar 09 '18 at 14:05

score -1 · Answer 2 · answered Mar 08 '18 at 16:02

If $E[X] < \infty$, then see here:

If $E[X] = \infty$, I guess you have to show

$$\lim_{i \to \infty} P(X \ge i) \ne 0 \leftarrow \lim_{i \to \infty} iP(X = i) \ne 0$$

Finding the expected value of a nonnegative integer valued random variable

2 Answers2