 Solution

ID: 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/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.

 New to topics? Read the docs here!