expectation

数学期望

Probability · Undergraduate · Mathematics Resources editorial compilation (AI-assisted)

Other names / variants: expected value · mean

Translation evidence: Literature usage

Meaning in context

For an integrable real random variable XX, its expectation is E[X]=∫ΩX dPE[X]=\int_\Omega X\,dP; for nonnegative XX, an extended value +∞+\infty is allowed.

可积实随机变量关于概率测度的积分；对非负变量可允许结果为正无穷。

Example

If P(X=2)=1/4P(X=2)=1/4 and P(X=0)=3/4P(X=0)=3/4, then E[X]=1/2E[X]=1/2.

Usage and distinctions

A Cauchy variable has no expectation; its symmetric principal value is not an expectation.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

  1. Gregory F. Lawler — Notes on ProbabilitySection 2, expectation definition · Checked 2026-10-14

Translation sources

  1. 李东风 — 应用随机过程（北京大学）§1.3.2 数字特征，定义1.11(1) · Checked 2026-10-14
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