Figure 3.6. The gamma density has a wide variety of shapes depending upon the parameter v. Here I have shown three densities, with a — 1 and v — 0.8 (panel a), v — 1 (panel b) or v — 2 (panel c).

so that a logarithmic plot of variance versus mean will have a slope of 2 when the underlying random variable has a gamma density (but the converse is not true - see Dick (2004)). This is an example of a mean-variance power relationship; more details are provided in Connections.

Second, the coefficient of variation of A is directly found from Eq. (3.56) and Exercise 3.12:

so that larger values of v imply less relative variation in the distribution of the encounter rate. Let us thus conduct a thought experiment in which we hold the mean I = v/a constant (fixed from some other rule, for example), but allow v n (obviously, then a must increase too). What happens to the probability densityf(l)? The density is becoming more and more peaked around the mean, because the coefficient of variation is getting smaller and smaller. In other words,fl) will approach a delta-function centered at the mean I.

It has been a long, but worthwhile, detour.

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