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Independent and identically distributed random variables

important notion in probability and statistics

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Record originEnglish Wikipedia
Text licenseCC BY-SA 4.0
Source revisionJun 27, 2026
Entity authorityQ378542 ↗
Source-derived summary

In probability theory and statistics, a collection of random variables is independent and identically distributed (i.i.d., iid, or IID) if each random variable has the same probability distribution as the others and all are mutually independent. IID was first defined in statistics and finds application in many fields, such as data mining and signal processing.

Introduction

Statistics commonly deals with random samples. A random sample can be thought of as a set of objects that are chosen randomly. More formally, it is "a sequence of independent, identically distributed (IID) random data points."

In other words, the terms random sample and IID are synonymous. In statistics, "random sample" is the typical terminology, but in probability, it is more common to say "IID."

Identically distributed means that there are no overall trends — the distribution does not fluctuate and all items in the sample are taken from the same probability distribution.

Independent means that the sample items are all independent events. In other words, they are not connected to each other in any way; knowledge of the value of one variable gives no information about the value of the other and vice versa.

Application

Independent and identically distributed random variables are often used as an assumption, which tends to simplify the underlying mathematics. In practical applications of statistical modeling, however, this assumption may or may not be realistic.

Editorial summary

This brief starts where responsible research should: with the source description of “Independent and identically distributed random variables” as important notion in probability and statistics. Everything that follows is an evidence route, not borrowed authority.

Editorial reviewA concise reference frame for defining the subject, testing terminology and identifying the institution closest to the evidence. The current 226-word lead offers orientation but no explicit four-digit date, so chronology should not be assumed. The selected authority fields contribute no independent date. The account is most persuasive where Independent, identically and distributed can be independently traced.
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Why this record matters

The subject matters to the general reference register because the source frames it as important notion in probability and statistics. Its deeper value depends on whether names, dates, institutions and citations support that framing.

Evidence profile

Named sources, stable identifiers and responsible institutions provide the strongest route from overview to verifiable evidence. The source revision retrieved here is dated Jun 27, 2026. The linked authority identifier is Q378542. None of the 0 selected statements returned an explicit reference.

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Source & attribution

This entry incorporates text from “Independent and identically distributed random variables” on English Wikipedia. Contributors are listed in the page history. Text is available under the Creative Commons Attribution-ShareAlike 4.0 License. Selected authority identifiers and statements are retrieved from Wikidata under CC0; their references and qualifiers remain part of the verification path.