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

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!

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Again, pursuing on a previous question without saying so... –  Did Mar 3 '13 at 17:59
@Did: I added the link. Thanks! –  Ethan Mar 3 '13 at 18:03

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.
@Ethan: For a normal Poisson process, it is $exp(-\lambda t)$. In inhomogeneous process, the $\lambda$ varies with time as $\lambda(t)$. –  Bravo Mar 4 '13 at 14:46