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

Table of contents
- Solution iv

Solution

Computing the same expected value directly from the Poisson distribution instead gives

B (n) = E [X^{n}] = j = 0 \sum \infty j^{n} \frac{e ^{- 1}}{j !} = \frac{1}{e} j = 0 \sum \infty \frac{j ^{n}}{j !} .

For each fixed positive integer

n

, the ratio test shows that this sum converges. This identity is Dobiński's formula. The case

n = 0

also gives

B (0) = 1

if

0^{0}

is interpreted as one in this particular moment formula.

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