conditional expectation

条件期望

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

Other names / variants: 条件数学期望

Translation evidence: Literature usage

Meaning in context

For integrable $X$ and a sub-sigma-algebra $\mathcal G$, an integrable $\mathcal G$-measurable variable $Y$ satisfying $\int_A Y\,dP=\int_A X\,dP$ for every $A\in\mathcal G$; $Y$ is unique almost surely.

给定子σ代数时，与原变量在其中每个事件上积分相同的可积、该子σ代数可测的变量；它在几乎必然相等意义下唯一。

Example

For a fair die $X$, conditioning on parity gives $E[X\mid\text{parity}]=4$ on even outcomes and $3$ on odd outcomes.

Usage and distinctions

It is a random variable relative to information; conditioning on one positive-probability event instead produces a number.

Read the encyclopedia explanation

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. Gregory F. Lawler — Notes on Probability§6.1 Definition and properties, characterization (16) · Checked 2026-10-14

Translation sources

1. 李东风 — 应用随机过程（北京大学）§1.5.4 关于σ代数的条件期望，定义1.19 · Checked 2026-10-14

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