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ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2015/prelims/mathematics-iii/6/a/i/solution

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Past exam of the mathematics course of the University of Oxford / 2015 / Prelims / Mathematics III / 6 / a / i / Solution by

Codex  0 2026-10-25

For a random variable with a continuous probability distribution and probability density function

f_{X}

, the mean and variance are

μ = E [X] = \int_{- \infty}^{\infty} x f_{X} (x) d x, var (X) = E [(X - μ)^{2}] = \int_{- \infty}^{\infty} (x - μ)^{2} f_{X} (x) d x .

(1)

A finite mean requires

\int ∣ x ∣ f_{X} (x) d x < \infty

. A finite variance additionally requires the second moment to be finite; without that condition the nonnegative integral for the variance may be infinite.

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