6.
\begin{figure}[h]
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\caption{Figure 1}
\includegraphics[alt={},max width=\textwidth]{f7137ba8-5526-4107-bccd-047de235d7d1-5_392_407_281_852}
\end{figure}
Figure 1 shows a square of side \(t\) and area \(t ^ { 2 }\) 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 \(t ^ { 2 }\). It is assumed that \(X\) and \(Y\) are independent random variables each having a continuous uniform distribution over the interval \([ 0 , t ]\).
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[You may assume that \(\mathrm { E } \left( X ^ { n } Y ^ { n } \right) = \mathrm { E } \left( X ^ { n } \right) \mathrm { E } \left( Y ^ { n } \right)\), where \(n\) is a positive integer.]
- Use integration to show that \(\mathrm { E } \left( X ^ { n } \right) = \frac { t ^ { n } } { n + 1 }\).
The random variable \(S = k X Y\), where \(k\) is a constant, is an unbiased estimator for \(t ^ { 2 }\).
- Find the value of \(k\).
- Show that \(\operatorname { Var } S = \frac { 7 t ^ { 4 } } { 9 }\).
The random variable \(U = q \left( X ^ { 2 } + Y ^ { 2 } \right)\), where \(q\) is a constant, is also an unbiased estimator for \(t ^ { 2 }\).
- Show that the value of \(q = \frac { 3 } { 2 }\).
- Find Var \(U\).
- State, giving a reason, which of \(S\) and \(U\) is the better estimator of \(t ^ { 2 }\).
The point \(( 2,3 )\) is selected from inside the square.
- Use the estimator chosen in part (f) to find an estimate for the area of the square.