Find the Mean for Non-Negative Integer-Valued Random Variable

Solution 1:

you could also proof this using telescoping series: $\begin{align*} \sum_{x=0}^\infty xP(X=x)&=\sum_{x=0}^\infty x(P(X>x-1)-P(X>x))\\ &=\sum_{x=0}^\infty xP(X>x-1)-\sum_{x=0}^\infty xP(X>x)\\ &= \sum_{x=1}^\infty xP(X>x-1)-\sum_{x=1}^\infty (x-1)P(X>x-1)\\ &=\sum_{x=1}^\infty (x-(x-1))P(X>x-1)\\ &=\sum_{x=1}^\infty P(X>x-1)\\ &=\sum_{x=0}^\infty P(X>x) \end{align*}$

Solution 2:

\begin{array} & & 0P(X=0) & + & 1P(X=1) & + & 2 P(X=2) & + & 3P(X=3) & + & \cdots \\[18pt] = & & & P(X=1) & + & P(X=2) & + & P(X=3) & + & \cdots \\ & & & & + & P(X=2) & + & P(X=3) & + & \cdots \\ & & & & & & + & P(X=3) & + & \cdots\\ & & & & & & & & + & \cdots \end{array}

The sum in the first row is $P(X>0)$; that in the second row is $P(X>1)$; that in the third row is $P(X>2)$, and so on.

Solution 3:

$\sum_{n=0}^{\infty}P(X>n)=\sum_{n=1}^{\infty}\sum_{x=n+1}^{\infty}P(X=x)=\sum_{x=0}^{\infty}\sum_{n=0}^{x-1}P(X=x)=\sum_{x=0}^{\infty}xP(X=x)=EX$

The interchange of the infinite sums is justified since $X$ has finite mean.