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.
Best Answer
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 $$ \lambda(t)=\left\{\begin{array}{ccc}\frac{f(t)}{1-F(t)}&\mathrm{if}&F(t)\lt1,\\0&\mathrm{if}&F(t)=1.\end{array}\right. $$