Past exam of the mathematics course of the University of Oxford / 2015 / Prelims / Mathematics III / 5 / a / iv

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- Solution iv

Solution

For the specified Poisson distribution, the exponential series gives the probability generating function

G (s) = e^{- 1} \sum_{j = 0}^{\infty} \frac{s ^{j}}{j !} = e^{s - 1}, s \in R .

Every derivative of

G

is again

e^{s - 1}

, so the factorial moments and variance are

E [(X)_{k}] = 1 (k \geq 1), var (X) = 1 + 1 - 1 = 1.

Also

E [X] = 1

.

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