Hello, fellow Wilmotters. Is someone familiar with reliability analysis? If so, can you help me give a proof of the following equation:For t >= 0F should have a bar on top, meaning that F bar = 1 - F(t)any ideas?thanks!

I will make an informed guess. Suppose we model the failure of something (say a device or the death of a person) asa Poisson process with a time-dependent intensity r(t). Roughly, this means that at time t the prob of surving forthe next Delta_t is 1 - r(t) (Delta_t) + O(Delta_t^2). Also, it means that the probability of surving for the next instantis independent of having survived so far. Given this independence assumption, the probability of surving over (0,T) is P(survival) = [1 - r(t_1) (Delta_t)] x [ 1 - r(t_2) (Delta_t)] x ... [1 - r(t_N) (Delta_t] + O((Delta_t)^2)where Delta_t = T/N, and the {t_i} are a partition of (0,T).Now just exponentiate, use that log (1 - r(t_i) Delta_t ) = - r(t_i) Delta_t + O((Delta_t)^2), and collect together all the leading terms. You should be able to recognize that the leading term in the exponential is a sum that tends to an integral in the limt N -> infinity, Delta_t -> 0, N Delta_t = T. You should also be able to show that the remainingterms vanish in this limit. The result is that: P(survival) = exp(- int_0^T r(t) dt)regards,

Thanks a lot, Alan. I did not intend to relate it to the Poisson Process but your explanation follows smoothly.Blade Runner rules!

Hello,The hazard function, r, is defined as:r(t)=\lim \Delta t \rightarrow 0 \frac{P(t\leq T <t+\Delta t|T\geq t)}{\Delta t}and the pdf function, f, f(t)=\lim \Delta t \rightarrow 0 \frac{P(t\leq T <t+\Delta t)}{\Delta t}Another formula expresses the hazard function as r(t)=\frac{f(t)}{S(t)}, where S(t)=1-F(t).Together, equations above imply that r(t)=-\frac{d}{dt}log(S(t))Integrating both sides of the last equation...