Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I am self-studying renewal processes and I came across an interesting problem. How would one go about finding the expectation of excess life given that current life is equal to x? I am assuming in this problem that the renewal process has an interarrival distribution function of F(x).

i.e. my problem is: What is E[excess life | current life = x]?

share|cite|improve this question
up vote 0 down vote accepted

$$ \frac1{1-F(x)}\int_0^{+\infty}tf(x+t)\mathrm dt. $$ A refined result is that, calling $C_y$ and $L_y$ the current life and the excess life at time $y$, for every $0\leqslant x\leqslant y$ the distribution of $L_y$ conditionally on $C_y=x$ is independent on $y$ and has density $g_x$, where $$ g_x(t)=\frac{f(x+t)\cdot[t\gt0]}{1-F(x)}. $$

share|cite|improve this answer
Can you please expand on this derivation? – icobes Mar 5 '12 at 22:45

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.