 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2019/discrete-mathematics-continuous-mathematics-and-probability-resit/9/a/v/solution

This is the ordinary factorial of

X

E [X!] = \sum_{k = 0}^{\infty} k! e^{- λ} \frac{λ ^{k}}{k !} = e^{- λ} \sum_{k = 0}^{\infty} λ^{k} .

The geometric series converges exactly when

0 < λ < 1

. Therefore $E [X!] = \frac{e ^{- λ}}{1 - λ}$ for $0 < λ < 1$ . For

λ \geq 1

, its terms fail to tend to zero and the expectation is infinite. This is the factorial of a Poisson random variable.

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