Q $1.$ Consider the exponential family of distribution in which the probability density$/$mass function takes the form

$f(y$;$,)=$exp$\{\frac{y-b()}{a}()+c(y,)\}.$

The probability density function of the gamma distribution is

$f(y$;$,)=\frac{1}{(\frac{1}{})}(\frac{y}{}{)}^{\frac{1}{}}\frac{1}{y}$exp$(-\frac{y}{}),y>0.$

$($a$)$ Show that the gamma distribution belongs to exponential family, and also identify $,,b(),a(),$ and $c(y,).$

$(8$ marks$)$

$($b$)$ Find the mean and variance of a gamma random variable in terms of $$ and $.$

$(4$ marks$)$

$($c$)$ Find the canonical link function.

$(2$ marks$)$

$($d$)$ Determine whether the following functions are suitable to be used as the link functions in gamma regression, and motivate your answer.

$($i$)g()=|-2|$

$(3$ marks$)$

$($ii$)g()=\frac{1}{}$

$(3$ marks$)$

$($iii$)g()=log()$

$(3$ marks$)$