Let Z = X + Y where X \sim N\left(\mu, \sigma^2 \right) and Y \sim \Gamma\left(k, \theta \right) using this parametrization of the Gamma distribution. Also assume X and Y are independent. Then what is the distribution (pdf) of Z?


This question doesn't seem as straightforward as it sounds. For example, I have tried using the convolution formulas here, but can't seem to find a closed form expression for the integral. I have also tried multiplying the moment generating functions (mgfs) of X and Y, but it does not seem to match up to any known mgfs.

Does anyone have any ideas on how to find the distribution for Z?