Past exam of the year 1 of the computer science course of the University of Oxford / 2017 / Discrete mathematics, continuous mathematics and probability / 9 / b

Table of contents
- i b
  - Solution i
- ii b
  - Solution ii

i

Solution

Assume finite second moments and write

Z_{i} = X_{i} - E X_{i}

. Expanding the squared centered sum gives

Var (X_{1} + X_{2}) = E [(Z_{1} + Z_{2})^{2}] = E [Z_{1}^{2}] + E [Z_{2}^{2}] + 2 E [Z_{1} Z_{2}] = Var (X_{1}) + Var (X_{2}) + 2 Cov (X_{1}, X_{2}) .

This variance of a sum identity does not require independence; the covariance term records the contribution that would otherwise be lost.

ii

Solution

For finite second moments, the general variance of a sum formula is

Var (i = 1 \sum n X_{i}) = i = 1 \sum n Var (X_{i}) + 2 1 \leq i < j \leq n \sum Cov (X_{i}, X_{j}) .

Each distinct unordered pair appears once in the second sum and twice when the full square is expanded. No independence assumption is needed.

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