overdispersion and hence is called the overdispersion parameter (eschewing the recommendation of Strunk and White about noun adjectives). Earlier, we concluded that if via is held fixed, but that v then the probability density for l will converge to a delta function centered on the mean I. Equation (3.61) is telling us the same information: that the overdispersion parameter increases, the mixture of Poisson distributions becomes more and more concentrated at a single value of the rate parameter and so the mean of the negative binomial distribution approaches the mean of the appropriate Poisson process. Indeed, as an optional (H) exercise, some readers may wish to show that in the limit of vn, Eq. (3.60) becomes the Poisson distribution.

Although I like to use the parameters v and a, there are other forms commonly used in the ecological literature. Perhaps the most common is the "m, k" form that gained considerable popularity through the seminal book of Sir Richard Southwood (Southwood 1978). For reasons that will become clear momentarily, let us start using the random variable N for the number of events and rewrite Eq. (3.60) as

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