# Gamma distribution from the sum of independent exponential distributions

From a paper I'm currently reading:

In the simplest setting, the average waiting time (or equivalently the departure rate) in each stage is assumed to be equal: the overall infectious period is then described by the sum of n independent exponential distributions, i.e. infectious periods are gamma distributed with a shape parameter n.

Essentially the idea is by having a waiting time go through n exponentially distributed steps, one can arrive at an overall waiting time that's gamma distributed with a shape parameter n.

That I understand - my question is, what's the scale parameter for that gamma distribution?

-
You don't get a gamma distribution unless the exponential random variables have the same scale parameter, and the gamma distribution inherits this common scale parameter. In the paper you are reading, the commonality of the scale parameter is guaranteed by the assumption that the average waiting time is the same in all the stages. – Dilip Sarwate Jan 13 '13 at 21:50

According to Forbes et al's Statistical Distributions 4th Edition page 111 this property of the sum of waiting times being gamma applies to all gamma distributions that share the same scale parameter $c$. To quote the above source, $\sum_{i=1}^n (\gamma : b, c_i) \sim \gamma: b, c$ where $c = \sum_{i=1}^n c_i$
Did you mean to make $b$ the shape parameter and $c$ the scale parameter? In your notation the gamma distributions appear to have a common value $b$ for one parameter and possibly distinct $c_i$ for the other. Judging from the Comment on the Question, you would want equal scale parameters and the shape parameter to be the sum of the stages' shape parameter. – hardmath Mar 20 '15 at 21:11