Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 2 / 28J / a

Solution

For

X_{1} \sim Poisson (θ)

G (s) = exp (θ (s - 1)) .

Thus

E [X_{1}] = G^{'} (1) = θ

and

Var (X_{1}) = G^{''} (1) + G^{'} (1) - G^{'} (1)^{2} = θ .

ℓ (θ) = - n θ + (\sum_{i = 1}^{n} X_{i}) lo g θ,

θ_{MLE} = \overline{X}

n (θ_{MLE} - θ) d N (0, θ) .

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