2. The random variable \(X\) follows a continuous uniform distribution over the interval \([ \alpha - 3,2 \alpha + 3 ]\) where \(\alpha\) is a constant.
The mean of a random sample of size \(n\) is denoted by \(\bar { X }\)
- Show that \(\bar { X }\) is a biased estimator of \(\alpha\), and state the bias.
Given that \(Y = k \bar { X }\) is an unbiased estimator for \(\alpha\)
- find the value of \(k\).
A random sample of 10 values of \(X\) is taken and the results are as follows
$$\begin{array} { l l l l l l l l l l }
3 & 5 & 8 & 12 & 4 & 13 & 10 & 8 & 5 & 12
\end{array}$$
- Hence estimate the maximum value of \(X\)