- The continuous random variable \(X\) has probability density function \(\mathrm { f } ( x )\)
$$f ( x ) = \left\{ \begin{array} { c c }
\frac { x } { 2 \theta ^ { 2 } } & 0 \leqslant x \leqslant 2 \theta
0 & \text { otherwise }
\end{array} \right.$$
where \(\theta\) is a constant.
- Use integration to show that \(\mathrm { E } \left( X ^ { N } \right) = \frac { 2 ^ { N + 1 } } { N + 2 } \theta ^ { N }\)
- Hence
- write down an expression for \(\mathrm { E } ( X )\) in terms of \(\theta\)
- find \(\operatorname { Var } ( X )\) in terms of \(\theta\)
A random sample \(X _ { 1 } , X _ { 2 } , \ldots , X _ { n }\) where \(n \geqslant 2\) is taken to estimate the value of \(\theta\) The random variable \(S _ { 1 } = q \bar { X }\) is an unbiased estimator of \(\theta\)
- Write down the value of \(q\) and show that \(S _ { 1 }\) is a consistent estimator of \(\theta\)
The continuous random variable \(Y\) is independent of \(X\) and is uniformly distributed over the interval \(\left[ 0 , \frac { 2 \theta } { 3 } \right]\), where \(\theta\) is the same unknown constant as in \(\mathrm { f } ( x )\).
The random variable \(S _ { 2 } = a X + b Y\) is an unbiased estimator of \(\theta\) and is based on one observation of \(X\) and one observation of \(Y\).
- Find the value of \(a\) and the value of \(b\) for which \(S _ { 2 }\) has minimum variance.
- Show that the minimum variance of \(S _ { 2 }\) is \(\frac { \theta ^ { 2 } } { 11 }\)
- Explain which of \(S _ { 1 }\) or \(S _ { 2 }\) is the better estimator for \(\theta\)