- A random sample \(X _ { 1 } , X _ { 2 } , \ldots , X _ { n }\) is taken from a population where each of the \(X _ { i }\) have a continuous uniform distribution over the interval \([ 0 , \beta ]\).
The random variable \(Y = \max \left\{ X _ { 1 } , X _ { 2 } , \ldots , X _ { n } \right\}\).
The probability density function of \(Y\) is given by
$$f ( y ) = \left\{ \begin{array} { c c }
\frac { n } { \beta ^ { n } } y ^ { n - 1 } & 0 \leqslant y \leqslant \beta
0 & \text { otherwise }
\end{array} \right.$$
- Show that \(\mathrm { E } \left( Y ^ { m } \right) = \frac { n } { n + m } \beta ^ { m }\).
- Write down \(\mathrm { E } ( Y )\).
- Using your answers to parts (a) and (b), or otherwise, show that
$$\operatorname { Var } ( Y ) = \frac { n } { ( n + 1 ) ^ { 2 } ( n + 2 ) } \beta ^ { 2 }$$
- State, giving your reasons, whether or not \(Y\) is a consistent estimator of \(\beta\).
The random variables \(M = 2 \bar { X }\), where \(\bar { X } = \frac { 1 } { n } \left( X _ { 1 } + X _ { 2 } + \ldots + X _ { n } \right)\), and \(S = k Y\), where \(k\) is a constant, are both unbiased estimators of \(\beta\).
- Find the value of \(k\) in terms of \(n\).
- State, giving your reasons, which of \(M\) and \(S\) is the better estimator of \(\beta\) in this case.
Five observations of \(X\) are: \(\quad \begin{array} { l l l l l } 8.5 & 6.3 & 5.4 & 9.1 & 7.6 \end{array}\)
- Calculate the better estimate of \(\beta\).