Show that \(U = a{\bar X_1} + (1 - a){\bar X_2},{\text{ }}a \in \mathbb{R}\), is an unbiased estimator of \(\mu \).

Show that \({\text{Var}}(U) = {a^2}\frac{{{\sigma ^2}}}{{{n_1}}} + {(1 - a)^2}\frac{{{\sigma ^2}}}{{{n_2}}}\).

Find, in terms of \({n_1}\) and \({n_2}\), an expression for \(a\) which gives the most efficient estimator of this form.

Hence find an expression for the most efficient estimator and interpret the result.

\({\text{E}}(U) = E(a{\bar X_1} + (1 - a){\bar X_2}) = a{\text{E}}({\bar X_1}) + (1 - a){\text{E}}({\bar X_2})\)     (M1)

\({\text{E}}({\bar X_1}) = \mu \) and \({\text{E}}({\bar X_2}) = \mu \)

\({\text{E}}(U) = a\mu  + (1 - a)\mu \) (or equivalent)      A1

\( = \mu \)     A1

hence \(U\) is an unbiased estimator of \(\mu \)     AG

[3 marks]

\({\text{Var}}(U) = {\text{Var}}(a{\bar X_1} + (1 - a){\bar X_2})\)

\( = {a^2}{\text{Var}}({\bar X_1}) + {(1 - a)^2}{\text{Var}}({\bar X_2})\)     M1

stating that \({\text{Var}}({\bar X_1}) = \frac{{{\sigma ^2}}}{{{n_1}}}\) and \({\text{Var}}({\bar X_2}) = \frac{{{\sigma ^2}}}{{{n_2}}}\)     A1

\( \Rightarrow {\text{Var}}(U) = {a^2}\frac{{{\sigma ^2}}}{{{n_1}}} + {(1 - a)^2}\frac{{{\sigma ^2}}}{{{n_2}}}\)     AG

Note:     Line 3 or equivalent must be seen somewhere.

[2 marks]

let \({\text{Var}}(U) = V\)

EITHER

\(\frac{{{\text{d}}V}}{{{\text{d}}a}} = 2a\frac{{{\sigma ^2}}}{{{n_1}}} - 2(1 - a)\frac{{{\sigma ^2}}}{{{n_2}}}\)     M1

attempting to solve \(\frac{{{\text{d}}V}}{{{\text{d}}a}} = 0\) for \(a\)     R1

Note:     Award M1 for obtaining \(a\) in terms of \({n_1},{\text{ }}{n_2}\) and \(\sigma \).

OR

forming a quadratic in \(a\)

\(V = \left( {\frac{{{\sigma ^2}}}{{{n_1}}} + \frac{{{\sigma ^2}}}{{{n_2}}}} \right){a^2} - 2\frac{{{\sigma ^2}}}{{{n_2}}}a + \frac{{{\sigma ^2}}}{{{n_2}}}\)     M1

attempting to find the axis of symmetry of V     R1

THEN

\(a = \frac{{\frac{{2{\sigma ^2}}}{{{n_2}}}}}{{2{\sigma ^2}\left( {\frac{1}{{{n_1}}} + \frac{1}{{{n_2}}}} \right)}}\)     (A1)

\(a = \frac{{{n_1}}}{{{n_1} + {n_2}}}\)     A1 

[4 marks]

substituting \(a\) into \(U\)     (M1)

\(U = \frac{{{n_1}{{\bar X}_1} + {n_2}{{\bar X}_2}}}{{{n_1} + {n_2}}}\)     A1

Note:     Do not FT an incorrect \(a\) for A1, the M1 may however be awarded.

this is an expression for the mean of the combined samples

OR this is a weighted mean of the two sample means     R1

[3 marks]