 Solution

ID: past-exam-of-the-options-ii-course-of-the-university-of-oxford/2009/m/m4/d/ii/solution

Write

Z_{t} = \sum_{j = 1}^{N_{t}} Y_{j}

, with an empty sum equal to zero. Conditioning on the independent Poisson random variable

N_{t}

E e^{θ Z_{t}} = \sum_{n = 0}^{\infty} e^{- λ t} \frac{( λ t ) ^{n}}{n !} M_{Y} (θ)^{n} = exp {λ t (M_{Y} (θ) - 1)} .

This is the compound Poisson distribution. Differentiating its log moment-generating function at zero, when the first two moments are finite, gives

E Z_{t} = λ t E Y, Var (Z_{t}) = λ t E (Y^{2}) .

The variance includes variability in the number of coaches, and so uses the second moment, not merely

Var (Y)

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