Survival Function

The survival function of an r.v. $X$ with CDF $F$ is the function $G$ given by $G(x)=1-F(x)=P(X>x)$

Let $X$ be a nonnegative r.v. Its expectation can be found by integrating its survival function:

$$\mathrm{E}[X]={\int }_0^{\infty }P(X>x)\mathrm{d}x$$

In the figure below, the area above a certain CDF and below the line $p=1$ is shaded. We can calculate the area of the shaded region by two ways

1. $\mathrm{Area}={\int }_0^1{F}^{-1}(p)\mathrm{d}p=\mathrm{E}\left[F^{-1}(U)\right]=\mathrm{E}[\mathrm{X}]$, where $U\sim \mathrm{Unif}(0,1)$
2. $\mathrm{Area}={\int }_0^{\infty }\left[1-F(x)\right]\mathrm{d}x=\mathrm{E}[X]$ More about $1$ can be found in the universality of the uniform distribution