91Ó°ÊÓ

If the random variable \(Y\) has the Gamma distribution with a scale parameter \(\beta,\) which is the parameter of interest, and a known shape parameter \(\alpha,\) then its probability density function is \\[ f(y ; \beta)=\frac{\beta^{\alpha}}{\Gamma(\alpha)} y^{\alpha-1} e^{-y \beta} \\] Show that this distribution belongs to the exponential family and find the natural parameter. Also using results in this chapter, find \(\mathrm{E}(Y)\) and \(\operatorname{var}(Y)\)

Is the extreme value (Gumbel) distribution, with probability density function $$f(y ; \theta)=\frac{1}{\phi} \exp \left\\{\frac{(y-\theta)}{\phi}-\exp \left[\frac{(y-\theta)}{\phi}\right]\right\\}$$ (where \(\phi>0\) is regarded as a nuisance parameter) a member of the exponential family?