6 In a certain city there are $N$ taxis. Each taxi displays a different licensing number which is an integer in the range 1 to $N$. A visitor to the city attempts to estimate the value of $N$, assuming that the licensing number of each taxi observed is equally likely to be any integer from 1 to $N$ inclusive.\\
(i) The visitor observes one randomly chosen licensing number, $X$. Using standard results for $\sum _ { r = 1 } ^ { n } r$ and $\sum _ { r = 1 } ^ { n } r ^ { 2 }$, show that $\mathrm { E } ( X ) = \frac { 1 } { 2 } ( N + 1 )$ and $\operatorname { Var } ( X ) = \frac { 1 } { 12 } \left( N ^ { 2 } - 1 \right)$.

The mean of 40 independent observations of $X$ is denoted by $A$.\\
(ii) Find an unbiased estimator $E _ { 1 }$ of $N$ based on $A$, and state the approximate distribution of $E _ { 1 }$, giving the value(s) of any parameter(s).\\
$B$ is another random variable based on a random sample of 40 independent observations of $X$. It is given that $\mathrm { E } ( B ) = \frac { 40 } { 27 } N$ and that $\operatorname { Var } ( B ) = \alpha N ^ { 2 }$ where $\alpha$ is a constant.\\
(iii) Find an unbiased estimator $E _ { 2 }$ of $N$ based on $B$, and determine the set of values of $\alpha$ for which $\operatorname { Var } \left( E _ { 2 } \right) > \operatorname { Var } \left( E _ { 1 } \right)$ for all values of $N$.

\hfill \mbox{\textit{Pre-U Pre-U 9795/2 2018 Q6}}