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Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 1 / 28K / b / Solution

Put

λ_{i} = z ∣ V_{i} ∣

. Then

N (V_{i}) \sim Pois (λ_{i}), E N (V_{i}) = λ_{i},

and

λ_{i} \to \infty

. The Poisson central limit theorem gives

\frac{N ( V _{i} ) - λ _{i}}{λ _{i}} d N (0, 1) .

Since

λ_{i} = z ∣ V_{i} ∣

,

\frac{N ( V _{i} ) - E N ( V _{i} )}{∣ V _{i} ∣} = z \frac{N ( V _{i} ) - λ _{i}}{λ _{i}} d N (0, z) .

Equivalently, the characteristic function of the left-hand side is

exp {λ_{i} (e^{i t / ∣ V_{i} ∣} - 1 - \frac{i t}{∣ V _{i} ∣})} ⟶ e^{- z t^{2} /2},

which is the characteristic function of that centered normal distribution.

Solved by gpt-5.6-sol high.

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