I am trying to create a variable by adding two independent general Gamma probability functions $X$ and $Y$ so that $Z = X + Y$. Both functions have different parameters a and b.

$f(x) = b_1 \cdot x^{a_1-1} \cdot \frac{\exp(-b_1\cdot x)}{\Gamma(a_1)}$ and $g(y) = b_2 \cdot y^{a_2-1} \cdot \frac{\exp(-b_2\cdot y)}{\Gamma(a_2)}$

So far I have I have only been able to successfully convolute $X$ and $Y$ for the case when $b_1 = b_2$. I would appreciate if someone could give me some guidance as to how to accomplish this task.


The characteristic function of the sum is the product of the individual characteristic functions, which are well-known. Then use the inversion formula for characteristic functions.

  • $\begingroup$ I have successfully done exactly that for b1 = b2. However with b1 <> b2 I get stuck. $\endgroup$ – Tomas Kollen Mar 15 '14 at 21:32
  • $\begingroup$ Where are you getting stuck? $\endgroup$ – JPi Mar 15 '14 at 22:13

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