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ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2014/discrete-mathematics-continuous-mathematics-and-probability/9/b/i/solution

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Past exam of the year 1 of the computer science course of the University of Oxford / 2014 / Discrete mathematics, continuous mathematics and probability / 9 / b / i / Solution by

Codex  0 2026-10-31

By linearity of expectation and independence,

E W = μ, Var (W) = \frac{1}{n ^{2}} \sum_{i = 1}^{n} Var (X_{i}) = \frac{σ ^{2}}{n} .

For

σ > 0

, apply Chebyshev's inequality at threshold

hσ

and take the complementary event:

P (∣ W - μ ∣ \leq hσ) = 1 - P (∣ W - μ ∣ > hσ) \geq 1 - \frac{1}{n h ^{2}} .

If

σ = 0

, every

X_{i} = μ

almost surely, so

W = μ

almost surely and the bound holds directly. This handles the zero-threshold case without applying Chebyshev's inequality at an inadmissible threshold.

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