S4 June 2006 Q6

6.

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 \([0, t]\).

[You may assume that \(\mathrm{E}(X^nY^n) = \mathrm{E}(X^n)\mathrm{E}(Y^n)\), where \(n\) is a positive integer.]

The random variable \(S = kXY\), where \(k\) is a constant, is an unbiased estimator for \(t^2\).

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

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