# 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