# distribution of the product of a poisson and a bolzmann

What is the distribution of the product of two variables for which each of them has its own distribution(specifically one poisson and one bolzmann)? I found on wikipedia that for the sum of the two you should use a convolution, but how is this for the product?

The situation for which I need this: i want to know the distribution of the total amount of energy of particles in a given time interval when the number of particles is poisson distributed and the energy of each particle is bolzmann distributed.

The situation you describe is not the product of a Poisson and a Boltzmann, because different particles have different energies. Instead, it's a compound Poisson distribution. If $X_i$ are independent random variables with your Boltzmann distribution, and $Y$ is independent of the $X_i$ and has a Poisson distribution, your random variable is $$Z = \sum_{i=1}^Y X_i$$ Note that $E[e^{sZ} | Y ] = \left(E[e^{sX}]\right)^Y$, so if $g_X(s) = E[e^{sX}]$ is the moment generating function of the $X_i$ and $g_Y(s) = E[e^{sY}]$ is the moment generating function of $Y$, the moment generating function of $Z$ is $$E[e^{sZ}] = E \left[\left(E[e^{sX}]\right)^Y\right] = g_Y(\log g_X(s)) = e^{\lambda (g_X(s)-1)}$$