 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/ii/paper-1/29l/c/i/solution

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

For

y = 0, 1, \dots

P_{θ} (Y_{i} = y) = e^{- θ y} (1 - e^{- θ}),

Y_{i}

is geometric with success probability

1 - e^{- θ}

n lo g (1 - e^{- θ}) - n θ \overline{Y} .

Its score equation gives

e^{- θ} = \frac{Y}{1 + Y},

and hence

θ_{n} = lo g (\frac{1 + Y}{Y}) .

When

\overline{Y} = 0

, the extended MLE is

+ \infty

; this event has probability tending to zero.

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