 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2009/paper-2/17/a/solution

p

μ_{X} = \int_{- \infty}^{\infty} x p (x) d x, Var X = \int_{- \infty}^{\infty} (x - μ_{X})^{2} p (x) d x .

The mean requires an integrable first absolute moment, and a finite variance requires a finite second moment. When these hold,

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

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