A random sample of size \(2 , X _ { 1 }\) and \(X _ { 2 }\), is taken from the random variable \(X\) which has a continuous uniform distribution over the interval \([ - a , 2 a ] , a > 0\)
Show that \(\bar { X } = \frac { X _ { 1 } + X _ { 2 } } { 2 }\) is a biased estimator of \(a\) and find the bias.
The random variable \(Y = k \bar { X }\) is an unbiased estimator of \(a\).
Write down the value of the constant \(k\).
Find \(\operatorname { Var } ( Y )\).
The random variable \(M\) is the maximum of \(X _ { 1 }\) and \(X _ { 2 }\)
The probability density function, \(m ( x )\), of \(M\) is given by
$$m ( x ) = \left\{ \begin{array} { c l }
\frac { 2 ( x + a ) } { 9 a ^ { 2 } } & - a \leqslant x \leqslant 2 a
0 & \text { otherwise }
\end{array} \right.$$
Show that \(M\) is an unbiased estimator of \(a\).
Given that \(\mathrm { E } \left( M ^ { 2 } \right) = \frac { 3 } { 2 } a ^ { 2 }\)
find \(\operatorname { Var } ( M )\).
State, giving a reason, whether you would use \(Y\) or \(M\) as an estimator of \(a\).
A random sample of two values of \(X\) are 5 and - 1