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ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2022/discrete-mathematics-continuous-mathematics-and-probability-resit/9/a/ii/solution

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Past exam of the year 1 of the computer science course of the University of Oxford / 2022 / Discrete mathematics, continuous mathematics and probability resit / 9 / a / ii / Solution by

Codex  0 2026-10-25

For a probability density function

f

, the finite expected value is

E [X] = \int_{- \infty}^{\infty} x f (x) d x, \int_{- \infty}^{\infty} ∣ x ∣ f (x) d x < \infty.

(1)

The absolute integrability condition makes the signed integral unambiguous. One can also allow an extended expectation when exactly one of the positive and negative parts has infinite integral, obtaining

+ \infty

- \infty

. If both parts have infinite integral, the expectation is undefined; subtracting two infinities is not a definition.

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