TITLE: Find the Mean for Non-Negative Integer-Valued Random Variable
QUESTION [8 upvotes]: Let $X$ be a non-negative integer-valued random variable with finite mean.
Show that
$$E(X)=\sum^\infty_{n=0}P(X>n)$$
This is the hint from my lecturer.
"Start with the definition $E(X)=\sum^\infty_{x=1}xP(X=x)$. Rewrite the series as double sum."
For my opinion. I think the double sum have the form of $\sum\sum f(x)$, but how to get this form? And how to continue?

REPLY [2 votes]: $\sum_{n=0}^{\infty}P(X>n)=\sum_{n=1}^{\infty}\sum_{x=n+1}^{\infty}P(X=x)=\sum_{x=0}^{\infty}\sum_{n=0}^{x-1}P(X=x)=\sum_{x=0}^{\infty}xP(X=x)=EX$
The interchange of the infinite sums is justified since $X$ has finite mean.