# What is the PDF sum of N random variables

I have N random variables: $$X_1,...,X_N$$ which are all independent. The PDF (probability density function) of each random variable $$f_{X_i}=e^{-a/(x^{2/b})}x^{-(4+b)/b}$$. What is the PDF $$f_S(x)$$ where $$S=X_1+...+X_N$$. Thank you in advance.

• This question requires the self-study tag. Expect hints but not complete answers. – Michael Chernick Sep 12 at 22:55
• I tried the moment generating function but I was stacked. I will appreciate any help. – Z.Chem Sep 12 at 23:01
• Presumably you mean "stuck" rather than "stacked". Your density doesn't integrate to 1 but the density of $x^{2/b}$ looks proportional to an inverse gamma (or $1/x$ looks proportional to a generalized gamma. How does this problem arise? Are you sure you need a closed form for the pdf here? – Glen_b Sep 13 at 1:28
• ...you're aware of the convolution method, right? – Sheridan Grant Sep 13 at 4:32
• It seems that there is no closed form for the PDF. Really I tried many methods (even the convolution) but always I didn't find solution. Did you think that I have to use an approximation? – Z.Chem Sep 13 at 20:59