 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/ia/paper-2/11f/d/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

Let

σ^{2} = Var (2^{X_{1}}) \leq ℓ 2^{b}

. If

σ = 0

the result is immediate. Otherwise the central limit theorem applies to

L

, while the displayed threshold is at least

α σ / n

. Hence

n \to \infty lim P (\frac{∣ L - ℓ ∣}{ℓ} \geq α \frac{2 ^{b}}{n ℓ}) \leq 2 - 2Φ (α) .

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