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Past exam of the mathematics course of the University of Cambridge / 2021 / ib / Paper 4 / 17H / a / Solution Created 2026-09-24 Updated 2026-09-25

Here

X

has the binomial distribution

Bin (n, θ)

, so, up to a factor independent of

θ

, the likelihood function is

L (θ; X) = θ^{X} (1 - θ)^{n - X} .

For

0 < X < n

, differentiating the log-likelihood gives

\frac{X}{θ} - \frac{n - X}{1 - θ} = 0,

and the boundary cases give the same formula. Thus the binomial proportion maximum-likelihood estimator is

θ = \frac{X}{n} .

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