A technician is trying to estimate the area $\mu^2$ of a metal square. The independent random variables $X_1$ and $X_2$ are each distributed $\text{N}(\mu, \sigma^2)$ and represent two measurements of the sides of the square. Two estimators of the area, $A_1$ and $A_2$, are proposed where

$$A_1 = X_1X_2 \text{ and } A_2 = \left(\frac{X_1 + X_2}{2}\right)^2.$$

[You may assume that if $X_1$ and $X_2$ are independent random variables then
$$\text{E}(X_1X_2) = \text{E}(X_1)\text{E}(X_2)$$]

\begin{enumerate}[label=(\alph*)]
\item Find $\text{E}(A_1)$ and show that $\text{E}(A_2) = \mu^2 + \frac{\sigma^2}{2}$.
[4]

\item Find the bias of each of these estimators.
[2]
\end{enumerate}

The technician is told that $\text{Var}(A_1) = \sigma^4 + 2\mu^2\sigma^2$ and $\text{Var}(A_2) = \frac{1}{2}\sigma^4 + 2\mu^2\sigma^2$. The technician decided to use $A_1$ as the estimator for $\mu^2$.

\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Suggest a possible reason for this decision.
[1]
\end{enumerate}

A statistician suggests taking a random sample of $n$ measurements of sides of the square and finding the mean $\overline{X}$. He knows that $\text{E}(\overline{X}^2) = \mu^2 + \frac{\sigma^2}{n}$ and

$$\text{Var}(\overline{X}^2) = \frac{2\sigma^4}{n^2} + \frac{4\sigma^2\mu^2}{n}.$$

\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{3}
\item Explain whether or not $\overline{X}^2$ is a consistent estimator of $\mu^2$.
[3]
\end{enumerate}

\hfill \mbox{\textit{Edexcel S4 2002 Q3 [10]}}