Gamma Distribution - Paul Johnson

Gamma Distribution Paul E. Johnson June 10, 2013

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Mathematical Description

Gamma is a probability model for a continuous variable on [0, ∞). It can look like a “ski slope” or it can look single-peaked. It has 2 parameters, shape and scale. I hate to put in symbols for these because I always forget which is which. So let’s just use the words. In some books, the scale is specified as 1/rate. Many models set scale=1, so rate=1, and there is no difference. If xi is Gamma distributed, the probability density function is: f (xi ) =

1 (shape−1) −( xi ) x e scale scaleshape Γ(shape) i

If you read a different book, you might find it rearranged like so: 1 xi f (xi ) = scale ∗ Γ(shape) scale 

(shape−1)

xi

e−( scale )

That’s really just the same distribution.

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Mathematical detail

What is that function Γ? R The function Γ(s) is the Gamma function, formally defined as Γ(s) = 0∞ ts−1 e−t dt if s > 0. The Gamma function arises in many statistical applications. The formula appears to be complicated, but just remember: its just the factorial function “extended” to take on values between the integers. Here’s why” If you pick s as an integer, Γ(s) is: Γ(s) = (s − 1)! f or s = 1, 2, 3... So, the value of Γ(1) = 1. And Γ(2) = 1, Γ(3) = 2, Γ(4) = 6, and so forth. And Γ(20) is some impossibly huge number. I never really worry very much about the value of Gamma, but in case you do, here’s a graph of it: 1

Figure 1: The Gamma Function x v a l s