Past exam of the mathematics course of the University of Cambridge / 2011 / ia / Paper 2 / 3F / a

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- Solution a

Solution

For a nonnegative integer-valued random variable, the probability generating function is

G_{X} (s) = E [s^{X}] = \sum_{n = 0}^{\infty} P (X = n) s^{n},

initially for

∣ s ∣ \leq 1

, with

0^{0} = 1

. For the Poisson distribution, insert its probability mass function and sum the exponential series:

G_{X} (s) = e^{- λ} \sum_{n = 0}^{\infty} \frac{( λ s ) ^{n}}{n !} .

Thus

G_{X} (s) = e^{λ (s - 1)} .

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