I have a continuous uniform random variable $P\sim U(0,1)$ and a normal random variable $X \sim N(0,\sigma)$. If $Z$ is given by: \begin{equation} Z = \frac{1}{1+P}+X \end{equation} where $X$ and $P$ are independent variables, how can I calculate the PDF of Z?
Any help anyone can give to guide me towards a solution would be greatly appreciated.
I've tried calculating the convolution of the PDFs with Mathematica as discussed here (https://mathematica.stackexchange.com/questions/245868/pdf-of-sum-of-1-1uniform-distribution-and-a-normal-distribution). Mathematica and (thanks to JimB!) Rubi both failed to find an explicit solution for the PDF of $Z$.