7 The continuous random variable \(X\) has probability density function
$$f ( x ) = \left\{ \begin{array} { c l }
\frac { k } { ( x + \theta ) ^ { 5 } } & \text { for } x \geqslant 0
0 & \text { otherwise }
\end{array} \right.$$
where \(k\) is a positive constant and \(\theta\) is a parameter taking positive values.
- Find an expression for \(k\) in terms of \(\theta\).
- Show that \(\mathrm { E } ( X ) = \frac { 1 } { 3 } \theta\).
You are given that \(\operatorname { Var } ( X ) = \frac { 2 } { 9 } \theta ^ { 2 }\). A random sample \(X _ { 1 } , X _ { 2 } , \ldots , X _ { n }\) of \(n\) observations of \(X\) is obtained. The estimator \(T _ { 1 }\) is defined as \(T _ { 1 } = \frac { 3 } { n } \sum _ { i = 1 } ^ { n } X _ { i }\).
- Show that \(T _ { 1 }\) is an unbiased estimator of \(\theta\), and find the variance of \(T _ { 1 }\).
- A second unbiased estimator \(T _ { 2 }\) is defined by \(T _ { 2 } = \frac { 1 } { 3 } \left( X _ { 1 } + 3 X _ { 2 } + 5 X _ { 3 } \right)\). For the case \(n = 3\), which of \(T _ { 1 }\) and \(T _ { 2 }\) is more efficient?
\section*{OCR}