$$

Let $U_1,U_2,$dots,$U_n$ be independent and identically distributed random variables from Uniform $(0,1).$

$($a$)$ Derive the probability density function of $x_1=-log{U}_1.$ Which family of distributions does it belong to$?$

$[5$ marks$]$

$($b$)$ Use the result in part $($a$),$ or otherwise, to show that the moment generating function of $Y=-log(U_1{U}_{2}cdots{U}_n)$ is

$M_Y(t)=\frac{1}{(1-t{)}^n},t<1,$

and hence determine the distribution of $Y.$

$[5$ marks$]$

$($c$)$ Determine the probability density function of $Z=U_1{U}_{2}cdots{U}_n.$

$[5$ marks$]$

$($d$)$ Express $E(Z^r)$ in terms of $n$ and $r.$

$[5$ marks$]$