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

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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

Inverse probability weighting for a causal mean Algorithm
Self-normalized importance sampling Algorithm
Standardization for a causal mean Algorithm
Holder's inequality for random variables Corollary
Layer-cake formulas for random variables Corollary
Linearity, monotonicity, and the modulus bound for expectation Corollary
Lyapunov central limit theorem 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
A supermartingale supremum threshold need not be attained Counterexample
Almost-sure convergence does not imply convergence of expectations Counterexample
An L1-bounded martingale need not converge in L1 Counterexample
An unbounded predictable transform may lose integrability Counterexample
Diminishing adaptation and uniform ergodicity need not give a strong law Counterexample
Doob Lp maximal inequality excludes p equals one Counterexample
Finite variance does not imply a sub-Gaussian tail Counterexample
Identical distribution without independence can defeat the mean law Counterexample
Integrable stopping time alone does not suffice for arbitrary martingale increments Counterexample
Quadratic pure-birth rates can explode Counterexample
Uniform moving support invalidates the naive score bound Counterexample
Uniform payoff convergence does not imply gradient convergence Counterexample
A metric for convergence in probability Definition
Adapted and integrable stochastic process Definition
Bias and mean squared error Definition
Binary instrumental-variable potential-outcome setup Definition
Binding function and indirect-inference target Definition
Biological and technical replication model Definition
Conditional expectation given a sigma algebra Definition
Contamination-robust expected risk and tail risk Definition
Continuous-time filtrations and all-pairs martingales Definition
Control-variate estimators Definition
Covariance matrix of a finite-dimensional random vector Definition
Critical, detection, and quantification levels Definition
Empirical, centered, and Rademacher processes Definition
False discovery proportion, FDR, and FWER Definition
Filtration-adapted stochastic-gradient oracles and Robbins–Monro updates Definition
Finite-dimensional linear mixed model with explicit longitudinal roles Definition
Fixed-design linear regression model Definition
Ideal and finite-simulation synthetic likelihood Definition

…and 123 more results.

Dependency tree · two levels

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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)