# Darth Vadar's Mean Survival Time

In this post we are interested in proving the Darth Vadar Rule:

where $S(x) = 1 - F(x)$ is the survival function. We also want to illustrate its usage in showing the closed form expression of mean survival time $E(t-t_0|t>t_0)$.

#### First, a Calculus proof of Darth Vadar Rule:

Now only need to show the claim $\lim_{x \rightarrow \infty} x*S(x) = 0$. We can show this by showing $0 \leq \lim_{x \rightarrow \infty} x*S(x) \leq 0$:

and conclude $\lim_{x \rightarrow \infty} x*S(x) = 0$ by the squeeze theorem.

#### Now, Mean Survival Time:

Want to show:

Note if we define $R = T-t_0$, above formula is just a straightforward application of the Darth Vadar rule, i.e.

we only need to express $S_R(r)$ in terms of $S_T(t)$:

and plug this expression into $\int_{\mathbb{R}_+} S_R(r) dr$: