Question

# The Gamma distribution is a distribution that arises naturally in processes for which the waiting times between events are relevant. It can be thought of as a waiting time between

Poisson distributed events, and is used, for example, to model the amount of rainfall accumulated in a reservoir, or the size of loan defaults or aggregate insurance claims. The gamma distribution is a two-parameter exponential family with two natural parameters (scale and shape). The density of a random variable Y with Gamma(a, v) distribution,a > 0, v, y < ∞, can be written as f_{Y}(y)=\frac{y^{\nu-1} \alpha^{\nu} e^{-y \alpha}}{\Gamma(\nu)} where I'(v) is a value of a known function that ensures that fy integrates to 1.We hold fixed v, the shape parameter. (a) Show that Gamma distribution with fixed v is a member of the one-parameter exponential family of distributions. [5 marks] (b) What are the canonical and dispersion parameters for the gamma distribution? Determine the variance function. [5 marks]

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