1 A parcel is weighed, independently, on two scales. The weights are given by the random variables $W _ { 1 }$ and $W _ { 2 }$ which have underlying Normal distributions as follows.

$$W _ { 1 } \sim \mathrm {~N} \left( \mu , \sigma _ { 1 } ^ { 2 } \right) , \quad W _ { 2 } \sim \mathrm {~N} \left( \mu , \sigma _ { 2 } ^ { 2 } \right) ,$$

where $\mu$ is an unknown parameter and $\sigma _ { 1 } ^ { 2 }$ and $\sigma _ { 2 } ^ { 2 }$ are taken as known.\\
(i) Show that the maximum likelihood estimator of $\mu$ is

$$\hat { \mu } = \frac { \sigma _ { 2 } ^ { 2 } } { \sigma _ { 1 } ^ { 2 } + \sigma _ { 2 } ^ { 2 } } W _ { 1 } + \frac { \sigma _ { 1 } ^ { 2 } } { \sigma _ { 1 } ^ { 2 } + \sigma _ { 2 } ^ { 2 } } W _ { 2 } .$$

[You may quote the probability density function of the general Normal distribution from page 9 in the MEI Examination Formulae and Tables Booklet (MF2).]\\
(ii) Show that $\hat { \mu }$ is an unbiased estimator of $\mu$.\\
(iii) Obtain the variance of $\hat { \mu }$.\\
(iv) A simpler estimator $T = \frac { 1 } { 2 } \left( W _ { 1 } + W _ { 2 } \right)$ is proposed. Write down the variance of $T$ and hence show that the relative efficiency of $T$ with respect to $\hat { \mu }$ is

$$y = \left( \frac { 2 \sigma _ { 1 } \sigma _ { 2 } } { \sigma _ { 1 } ^ { 2 } + \sigma _ { 2 } ^ { 2 } } \right) ^ { 2 }$$

(v) Show that $y \leqslant 1$ for all values of $\sigma _ { 1 } ^ { 2 }$ and $\sigma _ { 2 } ^ { 2 }$. Explain why this means that $\hat { \mu }$ is preferable to $T$ as an estimator of $\mu$.

\hfill \mbox{\textit{OCR MEI S4 2006 Q1 [24]}}