\includegraphics{figure_6}

Figure 1 shows a square of side 1 and area $l^2$ which lies in the first quadrant with one vertex at the origin. A point $P$ with coordinates $(X, Y)$ is selected at random inside the square and the coordinates are used to estimate $l^2$. It is assumed that $X$ and $Y$ are independent random variables each having a continuous uniform distribution over the interval $[0, l]$.

[You may assume that E$(X^n Y^m) = $ E$(X^n)$E$(Y^m)$, where $n$ is a positive integer.]

\begin{enumerate}[label=(\alph*)]
\item Use integration to show that E$(X^n) = \frac{l^{n+1}}{n+1}$. [3]
\end{enumerate}

The random variable $S = kXY$, where $k$ is a constant, is an unbiased estimator for $l^2$.

\begin{enumerate}[label=(\alph*)]
\item[(b)] Find the value of $k$. [3]

\item Show that Var $S = \frac{7l^4}{9}$. [3]
\end{enumerate}

The random variable $U = q(X^2 + Y^2)$, where $q$ is a constant, is also an unbiased estimator for $l^2$.

\begin{enumerate}[label=(\alph*)]
\item[(d)] Show that the value of $q = \frac{3}{2}$. [3]

\item Find Var $U$. [3]

\item State, giving a reason, which of $S$ and $U$ is the better estimator of $l^2$. [1]
\end{enumerate}

The point (2, 3) is selected from inside the square.

\begin{enumerate}[label=(\alph*)]
\item[(g)] Use the estimator chosen in part (f) to find an estimate for the area of the square. [1]
\end{enumerate}

TOTAL FOR PAPER: 75 MARKS

\hfill \mbox{\textit{Edexcel S4  Q6 [17]}}