What is the distribution of a random variable $U$ with $P(U⩾t)=\exp(−∫_0^t r(s)ds)$?

Question

From Did's comment following his reply, given a random variable $U$ with $P(U⩾t)=\exp(−∫_0^t r(s)ds)$ for some function $r:[0,\infty) \to [0, \infty)$ every $t⩾0$.

Is there a name for such a distribution?

If $r$ is constant, then $U$ has an exponential distribution.

If $r$ is piecewise constant, what is the name of the distribution? "Piecewise exponential"?

Thanks!

Again, pursuing on a previous question without saying so... – Did Mar 03 '13 at 17:59 — Did, Mar 03 '13 at 17:59

Bravo · Accepted Answer · 2013-03-03T19:15:45.643

1

I do not know if the distribution has a name, but it is the probability of the next arrival in an inhomogenous Poisson process taking time greater than $t$, i.e., $P(X>t)$ where $X$ is the inter-arrival time.

edited Mar 03 '13 at 19:15

answered Mar 03 '13 at 17:55

Bravo

4,413

Thanks! May I ask where you see the interarrival time for an inhomogeneous Poisson process has the distribution? – Tim Mar 04 '13 at 02:09
@Ethan: For a normal Poisson process, it is $exp(-\lambda t)$. In inhomogeneous process, the $\lambda$ varies with time as $\lambda(t)$. – Bravo Mar 04 '13 at 14:46

What is the distribution of a random variable $U$ with $P(U⩾t)=\exp(−∫_0^t r(s)ds)$?

1 Answers1