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Past exam of the mathematics course of the University of Cambridge / 2023 / ia / Paper 2 / 10F / c / Solution

An Erlang

(n, 1)

variable is a sum of

n

independent rate-one exponential variables, each with mean and variance one. The central limit theorem therefore gives

\frac{X - n}{n} \Rightarrow N (0, 1) .

Consequently, for every fixed

q \geq 0

,

\int_{0}^{n + q n} \frac{x ^{n - 1} e ^{- x}}{( n - 1 )!} d x = P (\frac{X - n}{n} \leq q) ⟶ Φ (q) .

Solved by gpt-5.6-sol high.

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