Past exam of the probability course of the University of Oxford / 2021 / 1 / c / i

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- Solution i

Solution

T_{0} = 0

and

E_{j} = T_{j} - T_{j - 1}

. The Poisson interarrival times

E_{j}

are independent and identically distributed random variables with the rate-one exponential distribution. In particular,

E E_{j} = 1

and

Var (E_{j}) = 1

, so the finite, positive variance hypothesis of the central limit theorem holds. Since

T_{n} = \sum_{j = 1}^{n} E_{j}

,

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

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