 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2017/prelims/mathematics-iii/6/b/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}) .

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