 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2022/discrete-mathematics-continuous-mathematics-and-probability/probability/7/a/i/solution

Assume finite second moments and set

Y_{i} = X_{i} - E X_{i}

X_{1} + X_{2}

gives

Y_{1} + Y_{2}

. Expanding its square and using the definitions of variance and covariance,

var (X_{1} + X_{2}) = E (Y_{1} + Y_{2})^{2} = var (X_{1}) + var (X_{2}) + 2 cov (X_{1}, X_{2}) .

No independence assumption is required for this variance of a sum identity.

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