tail sum formula

Tail Sum Formula

For a non-negative random variable $X\in \{0,\dots ,n\}$, the expected value can be computed using the tail sum formula:

$$\begin{array}{rl}\mathbb{E}[X]& =\sum _{k=1}^{n}P(X\ge k)\\ & =\sum _{k=0}^{n}kP(X=k)\\ & =1p_1+2p_2+3p_3+\cdots +(n-1)p_{n-1}+n{p}_n\\ & =\{\begin{array}{l}P(X\ge 1)\\ +P(X\ge 2)\\ +P(X\ge 3)\\ +\cdots \\ +P(X\ge n-1)\\ +P(X\ge n)\end{array}\begin{array}{l}=p_1+p_2+p_3+\cdots +p_n\\ +p_2+p_3+\cdots +p_n\\ +p_3+\cdots +p_n\\ +\cdots \\ +p_{n-1}+p_n\\ +p_n\end{array}\end{array}$$

Note that $P(X\ge k)=1-P(X\le k)=1-F_X(k)$, where $F_X$ is the cumulative distribution function of $X$.

Kind of reminds me of ${\sum }_{i=1}^{n}i=\frac{n(n+1)}{2}$.