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Past exam of the mathematics course of the University of Cambridge / 2021 / ib / Paper 4 / 17H / b / Solution

The estimator is unbiased because

E_{θ} X = n θ

, and

Var_{θ} (θ) = \frac{1}{n ^{2}} Var_{θ} (X) = \frac{θ ( 1 - θ )}{n} .

The bias-variance decomposition of mean squared error therefore gives

f (θ) = E_{θ} [(θ - θ)^{2}] = \frac{θ ( 1 - θ )}{n} .

The quadratic

θ (1 - θ)

is maximized at

θ = 1/2

, so

0 < θ < 1 sup f (θ) = \frac{1}{4 n} .

Solved by gpt-5.6-sol high.

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