>From: DanG3104@aol.com
>Subject: Sum of Gammas
>To: casnet@lists.casact.org
>
>I'm hoping someone on the list can help me with a probability problem. I
remember enough to frame the question, >but nowhere near enough to answer
it!
>
>I'm modeling a process made up of several independent, consecutive steps.
I'm using a Gamma distribution for the >duration of each step, so the total
duration is the sum of several independent Gamma variables. Is there a
closed >form for the distribution of the total duration, as a function of
the various Gamma parameters?
>
>I'd actually like to use a truncated Gamma (i.e., limited to a maximum and
minimum value) for the steps. Does that >make the distribution of the
total duration too messy?
>
>If there is no closed form, is there a reasonable approximation that
anyone can suggest? I'm doing this in Visual >Basic (Excel was way too
slow) so I am quite flexible about what kind of algorithm to implement.
>
>Thanks for your help!
>
>Dan Goddard
Dan, you might want to consider modeling your process steps with a Beta
distribution, instead of a Gamma. The Beta is a two-parameter
distribution, like the Gamma. But it's defined on [0,1] instead of all
positive X. So it's already truncated, and might be a better fit with the
underlying processes that you're trying to model than the Gamma.
In practical use, the Beta distribution is scaled and translated so that
it's defined for X on [Min, Max] instead of [0, 1]. When you consider the
two parameters of the distribution, plus the scale-factor and offset
(translation) factor you have a four-parameter distribution. For many
applications, the number of parameters can be reduced to three by making an
assumption about the shape of the distribution. The shape assumption
allows the distribution to (fairly) accurately match distributions such as
the normal and the exponential distributions. There are relatively simple
formulas for calculating the mean and the second and third moments about
the mean from the distribution's parameters and vice versa, which make the
scaled, translated Beta especially handy.
Regardless of whether you use the Gamma, Beta or some other distribution, a
good way to model the aggregate (total) distribution is to simply add up
the first three moments (the mean, variance and third moment about the
mean) for each individual distribution to get the moments of the total.
Then, if you have an easy way to calculate the parameters of the aggregate
distribution from the moments, you can come up with a "pretty good fit"
aggregate distribution. This approach works well with the Beta
distribution, and I've also seen it used with translated Gamma
distributions.
I've got details on exact calculations of Beta distributions for use in
spreadsheets, or some more approximate (but still darn good) calculations
for "quick and dirty" work that I use for project-length estimation. Let
me know if you'd like to see either/all of these approaches.
Mike Stouffer
Mike_Stouffer@aal.org
Visit the CAS Web Site at http://www.casact.org
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