Q$1.[15\%]$

a$)$ A random variable $Y,$ which is related to the angle $$ at which electrons are emitted in $-$meson decay, obeys a distribution with the probability density function

$(|)=\frac{1+y}{2},-1y1$ and $-11$

A random sample of experimental measurements $Y_1,Y_2,$dots.,$Y_n$ is collected and the following method of moments estimator $T$ for the parameter $$ is considered, where $T=\frac{3Y_1+3Y_2+\text{d}\text{o}\text{t}\text{s}\text{.}+3Y_n}{n}.$

$($i$)$ Determine whether $T$ is an unbiased estimator of the parameter $.$ Show all your working.

iil$)$ Determine whether $T$ is a consistent estimator of the parameter $.$ Show all your working.

$($iii$)$ Examine whether a fully efficient estimator of the parameter $$ could be found and give your conclusion.

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.