For a nonnegative integer-valued discrete random variable X, show that
E(X)=x=0 (to infinity) P(X>x). Can anyone help me solve this please?
So
$\displaystyle \sum_{x=0}^\infty P(X>x) = \sum_{x=0}^\infty \sum_{y=x+1}^\infty P(X=y)$
This is a triangular region. So we can switch the order of the sum if we do it carefully.
$\displaystyle = \sum_{y=1}^\infty \sum_{x=0}^{y-1} P(X=y)$
$\displaystyle = \sum_{y=1}^\infty y P(X=y)$
Can you finish it?
Well, there are no x's anymore. The x and y are just the summing variables.
The main observation at this point is that the sum:
$\displaystyle \sum_{y=1}^\infty yP(X=y) $
almost is $\displaystyle E[X]$ (whether it is x or y doesn't matter, it is just a summing variable). The only thing missing is the 0 term. The sum should go from 0 to $\displaystyle \infty$, whereas this one starts at 1. But the 0 term is
$\displaystyle 0P(X=0)=0$.
So $\displaystyle \sum_{y=1}^\infty yP(X=y) = 0P(X=0)+\sum_{y=1}^\infty yP(X=y) = \sum_{y=0}^\infty yP(X=y) =E[X]$