independent identically distributed

Two random variablesMathworldPlanetmath X and Y are said to be identically distributed if they are defined on the same probability spaceMathworldPlanetmath (Ω,,P), and the distribution functionMathworldPlanetmath FX of X and the distribution function FY of Y are the same: FX=FY. When X and Y are identically distributed, we write X=dY.

A set of random variables Xi, i in some index setMathworldPlanetmathPlanetmath I, is identically distributed if Xi=dXj for every pair i,jI.

A collectionMathworldPlanetmath of random variables Xi (iI) is said to be independent identically distributed, if the Xi’s are identically distributed, and mutually independentPlanetmathPlanetmath ( (every finite subfamily of Xi is independent). This is often abbreviated as iid.

For example, the interarrival times Ti of a Poisson process of rate λ are independent and each have an exponential distributionMathworldPlanetmath with mean 1/λ, so the Ti are independent identically distributed random variables.

Many other examples are found in statisticsMathworldMathworldPlanetmath, where individual data points are often assumed to realizations of iid random variables.

Title independent identically distributed
Canonical name IndependentIdenticallyDistributed
Date of creation 2013-03-22 14:27:29
Last modified on 2013-03-22 14:27:29
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 8
Author CWoo (3771)
Entry type Definition
Classification msc 60-00
Synonym iid
Synonym independent and identically distributed
Defines identically distributed