b$)x_1,x_2,$dots.,$x_n$ is a random sample from a uniform distribution $U(0,).$ The following

three statistics are being considered as estimators of the parameter $.$

$T_n=$ sample maximum, $,U_n=a{T}_n,V_n=b\stackrel{}{x},$

where $a,b$ are constants. You are given the following results.

$E(T_n)=\frac{n}{(n+1)},V(T_n)=\frac{n{}^2}{(n+1{)}^2(n+2)}$

and $U_n,V_n$ are unbiased estimators of $.$

$($i$)$ Find the values of the two constants a and b that will ensure both $U_n$

and $V_n$ are unbiased estimators of the parameter $.$ You must show all

your working.

$($ii$)$ Determine whether Un and $V_n$ are consistent estimators of $,$ giving

your reasons for the steps involved.