Past exam of the mathematics course of the University of Oxford / 2017 / Prelims / Mathematics III / 6 / 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 square in the definition of variance 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}) .

Recognizing the two variances and the covariance, we obtain

Var (X_{1} + X_{2}) = Var (X_{1}) + Var (X_{2}) + 2 Cov (X_{1}, X_{2}) .

ii

Solution

For random variables with finite second moments, the variance of a sum 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}) .

No independence assumption is needed.

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