This is a follow-up question from one I asked over at MathOverflow: https://mathoverflow.net/questions/158806/is-there-a-simple-closed-form-solution-for-the-joint-density-distribution-of-an
The previous question was: I have an exponential distribution with rate $\lambda$, where $\lambda$ is drawn from a Gamma distribution with shape and scale parameters $(k,\theta)$. I'd like to calculate an exact PDF for values, $v_i$, drawn from the exponential distribution if, for each sampling event, we randomly sample a value of $\lambda$ from the aforementioned Gamma distribution. Is there a simple closed-form solution for the PDF of the $v_i$?
My question here is: Is it possible to calculate a marginal distribution for the PDF of the $v_i$?