independent random variables

独立随机变量

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Other names / variants: independence of random variables

Translation evidence: Literature usage

Meaning in context

Variables XiX_i whose every finite subfamily satisfies P(Xij∈Bj, 1≤j≤r)=∏j=1rP(Xij∈Bj)P(X_{i_j}\in B_j,\ 1\le j\le r)=\prod_{j=1}^rP(X_{i_j}\in B_j) for all Borel sets BjB_j.

任意有限子族在任意指定Borel集合中同时取值的概率，都等于各自取值概率之积的随机变量族。

Example

The coordinates on a product of independent fair-coin spaces are independent random variables.

Usage and distinctions

Zero covariance is insufficient; independence concerns all measurable questions, not just first and second moments.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

  1. Gregory F. Lawler — Notes on ProbabilitySection 3, Independence, random-variable definition and infinite-family convention · Checked 2026-10-14

Translation sources

  1. 李东风 — 应用随机过程（北京大学）§1.5.1 独立性，定义1.15(5) · Checked 2026-10-14
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