Darth Vadar's Mean Survival Time

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

where is the survival function. We also want to illustrate its usage in showing the closed form expression of mean survival time .

First, a Calculus proof of Darth Vadar Rule:

Now only need to show the claim . We can show this by showing :

and conclude by the squeeze theorem.

Now, Mean Survival Time:

Want to show:

Note if we define , above formula is just a straightforward application of the Darth Vadar rule, i.e.

we only need to express in terms of :

and plug this expression into :