First, since there are a variety of parameterizations, I'll use Excel &
@Risk's f(x,a,b), where b=variance/mean and a=mean/b. (That's an easier way
to type out which parameter is which, and those two formulas are going to be
useful below.)
After I sent my message, I decided to fire up @Risk and see what a few
thousand simulations of a sum of Gammas might look like. It turns out to
look a lot like a Gamma. After a couple of guesses, I figured out that it
apparently is a Gamma whose mean is the sum of the component means (which is
what I expected), and whose variance is the sum of the variances. Using the
formulas above, you can then get a and b for the total distribution.
If the b's are the same, you get the result that a couple of my respondents
gave: just add the a's together and leave b as is.
However, not knowing any better, I had gone ahead and added Gammas with
different b values, which both respondents told me does not have a closed
form. But it sure looks like a Gamma to me. Generalized, I get the
following hypothesis:
If x1...xn are random variables with Gamma distributions with parameters
a1...an and b1...bn, then the sum of the x's has a Gamma distribution with
parameters
b = Sum of (ai bi^2) / Sum of (ai bi), and
a = Sum of (ai bi) / b
(The i's are supposed to be the subscripts that you're summing over)
For the more mathematically inclined (or those that Fred gives homework
assignments to), is this in fact true, or is it just a very close
approximation? If the latter, is it generally close, or just for the range
of parameters I happen to be using (a from 3 to 9, b from 50 to 200; what's
probably important is that b is large compared to a).
Of course, truncation messes up tidy results like this. I plan to do some
more testing to see how much.
Dan Goddard
Visit the CAS Web Site at http://www.casact.org
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