Past exam of the mathematics course of the University of Oxford / 2015 / Prelims / Mathematics III / 5 / b / iii

Table of contents
- Solution iii

Solution

Substitute

X

into the polynomial identity and apply linearity of expectation to its finite sum. The factorial moments of the Poisson distribution in part (a)(iv) are all one, giving

E [X^{n}] = k = 1 \sum n A (n, k) E [(X)_{k}] = k = 1 \sum n A (n, k) = B (n) .

Thus its

n

th moment counts set partitions: it equals the

n

th Bell number.

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