Q $5.$ Let $x_1,x_2,$cdots,$x_n$ be a random sample from a population with probability density function

$>0u(x_1,$cdots,$x_n)={\displaystyle \underset{i=1}{\overset{n}{}}}{x}_i\frac{1}{}f(x$;$)=\frac{1}{2}{}^3{x}^2{e}^{-x},0$

where $>0.$

$(a)$ Find the maximum likelihood estimator $of$

$(b)$ Show that $u(x_1,$cdots,$x_n)={\displaystyle \underset{i=1}{\overset{n}{}}}{x}_{i}is$ a sufficient statistic.

$(c)$ Show that the maximum likelihood estimator $in(b)isan$ unbiased estimator $of\frac{1}{}.$