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Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 2 / 12F / iv / Solution

If

N = 0

, then

∣ N - E N ∣ = E N

. Hence the Chebyshev inequality gives

P (N = 0) \leq P (∣ N - E N ∣ \geq E N) \leq \frac{var ( N )}{( E N ) ^{2}} .

The assumed limit therefore proves

P (N = 0) ⟶ 0 .

This is the second moment method in its simplest form.

Solved by gpt-5.6-sol high.

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