Past exam of the mathematics course of the University of Cambridge / 2022 / ib / Paper 2 / 6H / b

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- Solution b

Solution

The density is an equal mixture of the uniform distributions on

[0, 1]

and

[0, θ]

. Hence

E_{θ} X = \frac{1}{2} \cdot \frac{1}{2} + \frac{1}{2} \cdot \frac{θ}{2} = \frac{1 + θ}{4},

so

E_{θ} (4 \overline{X} - 1) = θ .

Thus

θ

is unbiased.

Also,

E_{θ} X^{2} = \frac{1 + θ ^{2}}{6}, Var_{θ} X = \frac{5 θ ^{2} - 6 θ + 5}{48} .

The central limit theorem gives

n (θ - θ) d N (0, \frac{5 θ ^{2} - 6 θ + 5}{3}) .

Replacing

θ

in the asymptotic standard error by the consistent estimator

θ

gives the Wald confidence interval

θ \pm z_{1 - α /2} \frac{5 θ ^{2} - 6 θ + 5}{3 n} .

It may be intersected with the parameter space

(0, 1)

.

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