You wrote:
>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?
The answer to this question is partially "yes". Let X(a,b) be a random variable with Gamma
distribution, i.e. its density function is equal to k(a,b) x^(a-1) exp(-bx), where
k(a,b)=b^a/G(a) is a normalizing constant, G(a) is Gamma function, "b" is a scaling
parameter, and "a" is a structural parameter. Then sum of two Gamma variables
X(a1,b)+X(a2,b) with equal values of scaling parameters is again Gamma variable
X(a1+a2,b), see Feller "An Introduction to Probability Theory and its Applications", 1971,
N.Y.:Wiley, vol. II, section II.2. However, if scaling parameters are different, the
distribution of X(a1,b1)+X(a2,b1) has no closed form. You may use numeric integration or
Monte Carlo simulation to approximate resulting distribution in the latter case.
>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 the underlying distributions are truncated Gamma, then there's no closed form for the
distribution of their sum. Numeric integration or Monte Carlo simulation might be used in
this case as well.
>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
>
Arcady.
=============================================
Arcady A. Novosyolov, Ph.D.
Institute of Computational Modeling, SB RAS,
19-149, Academgorodok, Krasnoyarsk, Russia, 660036,
tel. +7 3912 495382 (office), +7 3912 498596 (home)
fax (603) 688-4664 (U.S.A. number)
mailto: anov@cc.krascience.rssi.ru
Home page: http://www.geocities.com/novosyolov/
Forum: http://www.delphi.com/risktheory/
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