DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-04

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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Expectation of a nonnegative or integrable random variable

Definition

Let $(Ω, F, P)$ be a probability space.

If $X : Ω \to [0, + \infty]$ is measurable, its expectation is the extended-valued integral $E [X] : = \int_{Ω} X d P \in [0, + \infty] .$
If $X : Ω \to R$ or $X : Ω \to C$ is integrable in the sense of Integrable real and complex functions, and their integrals, its expectation is again $E [X] : = \int_{Ω} X d P,$ now a finite real or complex number.

For an integrable real random variable, $E [X] = E [X^{+}] - E [X^{-}],$ where $X^{+}$ and $X^{-}$ are its positive and negative parts.

Depends on

Used by

Holder's inequality for random variables Corollary
Layer-cake formulas for random variables Corollary
Linearity, monotonicity, and the modulus bound for expectation Corollary
Markov's inequality for random variables Corollary
The expectation of an indicator is the probability of the event Corollary
A random variable need not have a finite expectation Counterexample
Moments, variance, and covariance on a probability space Definition
The uniform random variable on [0,1] Example
Expectation depends only on the almost-everywhere class Lemma
Change of variables for expectation Theorem
Jensen's inequality for expectation Theorem

Dependency tree · two levels

11 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Rick Durrett, Probability: Theory and Examples, 5th ed., Sections 1.4 and 1.6 (standard reference, not scraped)
S. R. S. Varadhan, Probability Theory, Section 1.6 (standard reference, not scraped)