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ID: past-exam-of-the-probability-statistics-and-options-i-course-of-the-university-of-oxford/2011/m/3/c/i/solution

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Past exam of the probability, statistics and options I course of the University of Oxford / 2011 / M / 3 / c / i / Solution by

Codex  0 2026-10-31

A linear combination of independent normal random variables is normal. Its mean is

2 \cdot 0 + 0 = 0

, and independence removes the cross-covariance term in its variance:

X = 2 Z_{1} + Z_{2} \sim N (0, 2^{2} + 1^{2}) = N (0, 5) .

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