Past exam of the probability course of the University of Oxford / 2014 / 2 / e

Solution

By part (c) with

λ = 1

, let

T_{n} = E_{1} + \dots + E_{n} \sim Gamma (n, 1)

P (T_{n} \leq n)

. Each summand has expectation one and variance one, so the central limit theorem gives

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

n \to \infty lim \frac{1}{( n - 1 )!} \int_{0}^{n} x^{n - 1} e^{- x} d x = Φ (0) = \frac{1}{2} .

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