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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.

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  • $\begingroup$ This question requires the self-study tag. Expect hints but not complete answers. $\endgroup$ – Michael Chernick Sep 12 at 22:55
  • $\begingroup$ I tried the moment generating function but I was stacked. I will appreciate any help. $\endgroup$ – Z.Chem Sep 12 at 23:01
  • $\begingroup$ 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? $\endgroup$ – Glen_b Sep 13 at 1:28
  • $\begingroup$ ...you're aware of the convolution method, right? $\endgroup$ – Sheridan Grant Sep 13 at 4:32
  • $\begingroup$ 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? $\endgroup$ – Z.Chem Sep 13 at 20:59

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