# Exponential Distribution with changing (time-varying) rate parameter

In short, how would one sample an Exponential Distribution with an exponentially-increasing rate parameter, e.g., $\lambda=e^t$? What distribution does such a random variable follow?

Background: The Exponential Distribution models "time until failure" of, say, lightbulbs. It is parametrized by a constant parameter $\lambda$ called the failure rate that is the average rate of lightbulb burnouts. If $\lambda$ is time-varying, say, proportionally to a power of time, then it turns out that the time until failure is distributed by the more general Weibull distribution. This question involves this relationship, although the time-variance of $\lambda$ is not made explicit.

My question is: What if $\lambda$ varies exponentially in time? Is there a distribution for this case? More simply, how would I sample a random variable that is exponentially distributed with time-varying rate parameter $\lambda = e^t$?

I've searched Google and Wikipedia without success. I'd really appreciate any advice or pointers. Thanks in advance.

If $\lambda(t)=\mathrm e^t$, the density $f$ of the time until failure is such that, for every $t\geqslant0$, $$f(t)=\mathrm e^{t+1-\mathrm e^t}.$$ More generally, $$f(t)=\lambda(t)\,\mathrm e^{-\Lambda(t)},\quad\Lambda(t)=\int_0^t\lambda(s)\,\mathrm ds.$$ Note that every distribution with PDF $f$ and CDF $F$ can be written as such, using $$\lambda(t)=\left\{\begin{array}{ccc}\frac{f(t)}{1-F(t)}&\mathrm{if}&F(t)\lt1,\\0&\mathrm{if}&F(t)=1.\end{array}\right.$$

• Thanks for the response! But I am not clear on (1) proofs/references supporting these statements or (2) how they apply to my question. – ConvexMartian Mar 24 '14 at 16:04
• "Amusing" comment by the OP, which I had somehow forgotten... – Did Dec 7 '17 at 6:21

You can find proofs/references of what is stated above here.

To sample a value from a non-homogeneous exponential distribution you can follow this steps

S1. Sample $$x$$ from homogeneous exponential with rate 1

S2. Calculate $$\Lambda^{-1} (x)$$

where $$\Lambda(t)$$ is the intensity function (the integral of the rate).

The random variable $$\Lambda^{-1} (x)$$ has a non-homogeneous distribution with rate $$\lambda(t)$$.

A reference could be the paper "Generating Nonhomogeneous Poisson Processes" by Raghu Pasupathy.