# How to obtain the gamma distribution through convolution of two different distributions?

How can one obtain the gamma distribution through convolution of two different distributions? Could the gamma distribution be created as a non-trivial sum of $N$ random variables $X$ which have the same distribution and parameters?

Trivial case is summation of fixed number $N$ of variables with gamma distribution as described on Wikipedia.

• Your title and the beginning of your question references two different distributions, but later you talk about a sum of random variables from the same distribution. Which are you interested in? – cardinal May 7 '12 at 0:58
• This problem is solved for any positive integral $n$ by taking the $n^\text{th}$ root of the characteristic function of the given Gamma distribution. This yields a c.f. which--by inspection--is easily seen to correspond to (another) Gamma distribution (with the same scale parameter but a different shape parameter). – whuber May 7 '12 at 17:12