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Date May 2017 Marks available 3 Reference code 17M.3sp.hl.TZ0.4
Level HL only Paper Paper 3 Statistics and probability Time zone TZ0
Command term Hence and Find Question number 4 Adapted from N/A

Question

The random variables \({X_1}\) and \({X_2}\) are a random sample from \({\text{N}}(\mu ,{\text{ 2}}{\sigma ^2})\). The random variables \({Y_1}\), \({Y_2}\) and \({Y_3}\) are a random sample from \({\text{N}}(2\mu ,{\text{ }}{\sigma ^2})\).

The estimator \(U\) is used to estimate \(\mu \) where \(U = a({X_1} + {X_2}) + b({Y_1} + {Y_2} + {Y_3})\) and \(a\), \(b\) are constants.

Given that \(U\) is unbiased, show that \(2a + 6b = 1\).

[3]
a.

Show that \({\text{Var}}(U) = (39{b^2} - 12b + 1){\sigma ^2}\).

[3]
b.

Hence find the value of \(a\) and the value of \(b\) which give the best unbiased estimator of this form, giving your answers as fractions.

[3]
c.i.

Hence find the variance of this best unbiased estimator.

[1]
c.ii.

Markscheme

\({\text{E}}(U) = a\left( {{\text{E}}({X_1}) + {\text{E}}({X_2})} \right) + b\left( {{\text{E}}({Y_1}) + {\text{E}}({Y_2}) + {\text{E}}({Y_3})} \right)\)     (M1)

\( = 2a\mu  + 6b\mu \)     A1

(for an unbiased estimator,) \({\text{E}}(U) = \mu \)     R1

giving \(2a + 6b = 1\)     AG

 

Note:     Condone omission of E on LHS.

 

[3 marks]

a.

\({\text{Var}}(U) = {a^2}\left( {{\text{Var}}({X_1}) + {\text{Var}}({X_2})} \right) + {b^2}\left( {{\text{Var}}({Y_1}) + {\text{Var}}({Y_2}) + {\text{Var}}({Y_3})} \right)\)     (M1)

\( = 4{a^2}{\sigma ^2} + 3{b^2}{\sigma ^2}\)     A1

\( = 4{\left( {\frac{{1 - 6b}}{2}} \right)^2}{\sigma ^2} + 3{b^2}{\sigma ^2}\)     A1

\( = (39{b^2} - 12b + 1){\sigma ^2}\)     AG

[3 marks]

b.

the best unbiased estimator (of this form) will be found by minimising \({\text{Var}}(U)\)     (R1)

For example, \(\frac{{\text{d}}}{{{\text{d}}b}}\left( {{\text{Var}}(U)} \right) = (78b - 12){\sigma ^2}\)     (A1)

for a minimum, \(b = \frac{{12}}{{78}}\,\,\,\left( { = \frac{2}{{13}}} \right)\) so that \(a = \frac{3}{{78}}\,\,\,\left( { = \frac{1}{{26}}} \right)\)     A1

[3 marks]

c.i.

\({\text{Var}}U = \left( {39{{\left( {\frac{2}{{13}}} \right)}^2} - 12\left( {\frac{2}{{13}}} \right) + 1} \right){\sigma ^2}\)

\( = \frac{{{\sigma ^2}}}{{13}}\,\,\,(0.0769{\sigma ^2})\)     A1

[1 mark]

c.ii.

Examiners report

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Syllabus sections

Topic 7 - Option: Statistics and probability » 7.3

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