 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/ii/paper-1/5j/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

- \frac{( x - μ ) ^{2}}{2 μ ^{2} x} = - \frac{x}{2 μ ^{2}} + \frac{1}{μ} - \frac{1}{2 x} .

Consequently its probability density function can be written as

f (x; μ) = h (x) exp {η x - A (η)},

where

h (x) = \frac{1}{2 π x ^{3}} exp (- \frac{1}{2 x}), η = - \frac{1}{2 μ ^{2}} < 0, A (η) = - - 2 η .

Indeed,

- A (η) = - 2 η = 1/ μ

η = - \frac{1}{2 μ ^{2}} .

Here the natural statistic is

T (X) = X

E_{η} X = A^{'} (η) = (- 2 η)^{- 1/2} = μ

and

var_{η} X = A^{''} (η) = (- 2 η)^{- 3/2} = μ^{3} .

Thus

E X = μ, var X = μ^{3} .

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