# What reference can I cite for the proof that the sum of n exponential variables follows a gamma distribution?

There is a fairly common theorem, which states that:

The sum of $n$ independent variables following an exponential distribution $\mathrm{Exp}(\alpha)$ follow an gamma distribution $\mathrm{Gamma} (n, 1/\alpha)$ (also known as Erlang distribution).

I'm using this theorem in my thesis. I've been asked to cite the paper that introduces, or, if it's not possible, to at least cite a paper that explicitly mentions it.

Anyone knows of any such papers?

• Does it have to be a paper? This is a standard result found in many text books. I'd even suggest providing the proof yourself and placing it in an appendix. Oct 22, 2011 at 21:18
• See, for example, R. Durrett (2005), Probability: Theory and Examples, 3rd ed., Duxbury Press, page 30. The recent fourth edition is available on his website and you can search for the corresponding page number there. Oct 22, 2011 at 21:23
• Just be careful about notation. Some people write $\Gamma(n,\lambda)$ instead of $\Gamma(n, \frac{1}{\lambda})$ as you write it. In either case, the expected value is $\frac{n}{\lambda}$. And of course, the result is more general too: The sum of independent $\Gamma(s_i, *)$ random variables is a $\Gamma(\sum_i s_i, *)$ random variable ($*$ can be $\lambda$ or $\frac{1}{\lambda}$ whichever you like). See for example, Sheldon Ross's A First Course in Probability, Chapter 6. Oct 23, 2011 at 1:28
• Why create an appendix (@cardinal) where a short sentence will work? It suffices to point out that the characteristic function of $\Gamma(n,\alpha)$ is $(1-\alpha i t)^{-k}$ which is the $k^\text{th}$ power of $(1-\alpha i t)^{-1}$, the cf of the Exponential($\alpha$) distribution. This implicitly defines the notation, too (@Dilip).
– whuber
Oct 24, 2011 at 15:09
• @whuber: it seems to me we are on the same page here. My original remark was meant to suggest doing both since providing the proof explicitly would be more authoritative. Based on the committee's request I am guessing that this is not a thesis in statistics. So, that was another motivation for the suggestion regarding the appendix; that is, a proof may not seem relevant or necessary within the body of the thesis. Oct 24, 2011 at 15:34