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The Gamma(0, b, a) distribution returns the "time" we will have to wait before observing a independent Poisson events, where one has to wait on average b units of "time" between each event. The "time" to wait before a single event occurs is a Gamma(0,b,1) = Exponential(1/b) distribution, with mean b and standard deviation b too. The Gamma(0, b, a) is thus the sum of a independent Exponential(1/b) distributions, so Central Limit Theorem tells us for sufficiently large a (>30, for example), we can make the approximation:

 

Gamma(0, b, a) » Normal(\alpha \beta ,\sqrt{\alpha}\beta)

 

The Gamma(0, b, a) distribution has mean and standard deviation equal to ab and a½b respectively, which provides a nice check to our approximation.

 

 

 

 


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