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ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2015/prelims/mathematics-iii/4/c/i/solution

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Past exam of the mathematics course of the University of Oxford / 2015 / Prelims / Mathematics III / 4 / c / i / Solution by

Codex  0 2026-10-25

Each random variable has mean zero and second moment

2/3

, so its variance is

2/3

. For

S_{m} = \sum_{i = 1}^{m} X_{i}

, linearity of expectation and variance additivity for independent random variables give

E [S_{m}] = 0, var (S_{m}) = \frac{2 m}{3} .

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