# The gamma distribution and Poisson processes

I know that the gamma distribution with parameters $k$ and $\theta$ can be used as a model for the occurrence of events. The requirement on the events would be that their occurrence is random and the mean time between them equals $\theta$ and that they belong to a Poisson process.

Now I have a question regarding the applicability of these:

I want to model a biological growth process where we think of the resulting length as random. What we do know is that the underlying processes (biochemical etc.) are Poisson processes. Is it in the context of the statement above appropriate to model the length with a gamma random variable?

1. Interval between events in a Poisson process is exponentially distributed. If you skip intervals, and only count k-th event, then you get convolution of exponential r.v.'s which gives you Erlang distribution (which is a special case of Gamma distribution). Is this what you meant in the first paragraph?

2. If your growth process is such that it adds an iid Gamma distributed length every time, then you will get a Gamma distributed length per time (sums of Gamma r.v. with same scale is again Gamma). But this has little to do with Poisson process, I'm afraid.

• Thanks, but well no, my growth process is such that a gamma variable is added at each time unit..
– Pugl
Aug 21, 2013 at 17:15
• @Pegah sum of gamma r.v. is again a gamma r.v., so what is your question? You should clarify your question, I think. Aug 21, 2013 at 17:24
• So, let's say we have three interacting processes which lead together to growth. All these processes are poisson processes. I was told that in this case, I can model the growth of the length as a gamma variable since it "arises out of poisson processes", I found that a bit vague and was wondering if this modeling approach is justified..I am sorry I can not make it more precise.
– Pugl
Aug 21, 2013 at 18:31
• The Gamma distribution is conjugate to the Poisson distribution. Perhaps this is what was meant?
– Sycorax
Aug 21, 2013 at 19:31
• @Pegah What kind of interaction are we talking here? Do you have a reference? If there's a single Poisson process that generates a unit length growth, the total growth in some time would lead to a Poisson distribution which is NOT a Gamma distribution. If there are many Poisson processes summed, it's still a Poisson distribution. It may be close to a Gamma, but then again, it is close to Gaussian as well. Aug 21, 2013 at 19:32