Back to the gamma density

Now that we are more familiar with the gamma function, let us return to the gamma density given by Eq. (3.52). As with the gamma function, I will be as brief as possible, so that we can get back to the negative binomial distribution. In particular, we will examine the shape of the gamma density and find the mean and variance.

First, let us think about the shape of the gamma density (Figure 3.6). When v = 1, the algebraic term disappears and the gamma density is the same as the exponential distribution. When v > 1, the term 2v — pins f (0) = 0 so that the gamma density will rise and then fall. Finally, when v < 1, f (2) as 2 ! 0. We thus see that the gamma density has a wide variety of shapes.

If we let A denote the random variable that is the rate of the Poisson process, then

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