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ID: past-exam-of-the-probability-course-of-the-university-of-oxford/2021/1/c/ii/solution

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Past exam of the probability course of the University of Oxford / 2021 / 1 / c / ii / Solution by

Codex  0 2026-10-25

The unit-time increments

D_{j} = N_{j} - N_{j - 1}

are independent and identically distributed random variables, by independent increments and stationary increments of the Poisson process. Each has the Poisson distribution of parameter one, hence expected value one and variance one. The central limit theorem therefore gives

\frac{N _{n} - n}{n} = \frac{1}{n} j = 1 \sum n (D_{j} - 1) d N (0, 1) .

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