independent and identically distributed

独立同分布

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

Other names / variants: i.i.d. · iid

Translation evidence: Literature usage

Meaning in context

A family of random variables that is mutually independent and whose members all have the same probability law.

各成员具有相同分布且整个变量族相互独立。

Example

Independent repeated rolls of the same fair die give an i.i.d. sequence.

Usage and distinctions

Identical marginals do not imply independence: repeating one random variable at every time gives the same marginals but dependence.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

  1. Gregory F. Lawler — Notes on ProbabilitySection 4, i.i.d. definition before the weak law · Checked 2026-10-14

Translation sources

  1. 李东风 — 应用随机过程（北京大学）§1.5.3 例1.32，独立同分布序列 · Checked 2026-10-14
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