How to compute the limiting distribution of excess life, with uniform density? I'm working on the reward/renewal process question and get stuck in the following question.
could anyone please lend me some help?
--
Question:
What is the limiting distribution of excess life,
when renewal lifetimes have the uniform density $f(x) = 1$, for $0<x<1$.
I know the answer is $h(x)=2(1-x)$,   but I don't know how this answer came up.

We know the mean $\mu$ for a uniform distribution is $\frac{1}{2}$. So utilizing what we know from Ross: 
$$
\lim_{t \rightarrow \infty} P(\gamma_t \leq x) = \frac{1}{\mu} \int_0^x [1-F(t)] dt = 2x-x^2
$$ 
So the density is $2(1-x)$ for $0 < x <1$
