Past exam of the mathematics course of the University of Oxford / 2015 / Prelims / Mathematics III / 7 / b / iv

Table of contents
- Solution iv

Solution

The recurrence relation at

k = 3

gives

I_{5} = 2 θ I_{3} = 2 θ^{2}

. Consequently

var (X_{i}^{2}) = E [X_{i}^{4}] - (E [X_{i}^{2}])^{2} = I_{5} - I_{3}^{2} = 2 θ^{2} - θ^{2} = θ^{2} .

By variance additivity for independent random variables, the variance of their sample mean is

var (θ) = \frac{n θ ^{2}}{n ^{2}} = \frac{θ ^{2}}{n} .

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