Is there a simple closed form solution for the joint density distribution of an exponential distribution with a rate given by a Gamma distribution?

Question

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 $v_i$?

Answer 1

$$ f(t,s)= f(t\mid s)\cdot f(s) = \frac1{\theta^k\Gamma(k)} s e^{-st}\cdot s^{k-1}\cdot e^{-s/\theta} $$ $$ = \frac1{\theta^k\Gamma(k)} s^{k}\cdot e^{-st-s/\theta} $$ for $s>0$, $t>0$. Here $s=\lambda$.