 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2015/discrete-mathematics-continuous-mathematics-and-probability/9/a/ii/solution

Assume a finite second moment, so the following expectations are finite. Expand the square and use linearity of expectation:

var (X) = E [X^{2} - 2 μ X + μ^{2}] = E [X^{2}] - 2 μ E [X] + μ^{2} = E [X^{2}] - μ^{2} .

Thus

var (X) = E [X^{2}] - (E [X])^{2} .

The same equality holds in the extended sense when the mean is finite and the second moment is infinite.

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