 Solution

ID: past-exam-of-the-probability-course-of-the-university-of-oxford/2015/1/c/i/solution

N_{k} \sim Pois (k)

, with expected value and variance both equal to

k

. Consequently

Y_{k} = (N_{k} - k) / k

. Scaling and translating the argument in the moment-generating function of a Poisson distribution gives, for every real

t

M_{Y_{k}} (t) = e^{- t k} M_{N_{k}} (t / k) = exp {k (e^{t / k} - 1 - \frac{t}{k})} .

The centering term must remain inside the exponent: omitting it would leave a divergent first-order contribution.

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