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Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 1 / 29L / c / i / Solution

For

y = 0, 1, \dots

,

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

so

Y_{i}

is geometric with success probability

1 - e^{- θ}

. The log likelihood is

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.

Solved by gpt-5.6-sol high.

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