I.i.d.

In probability theory and statistics, a sequence or other collection of random variables is independent and identically distributed (i.i.d. or iid or IID) if each random variable has the same probability distribution as the others and all are mutually independent. Identically distributed, on its own, is often abbreviated ID. For uniformity, as both are discussed—and in widespread use—this article uses the visually cleaner IID in preference to the more prevalent convention i.i.d.

The annotation IID is particularly common in statistics , where observations in a sample are often assumed to be effectively IID for the purposes of statistical inference. The assumption (or requirement) that observations be IID tends to simplify the underlying mathematics of many statistical methods (see mathematical statistics and statistical theory). However, in practical applications of statistical modeling the assumption may or may not be realistic. To test how realistic the assumption is on a given data set, the can be computed, lag plots drawn or turning point test performed. The generalization of exchangeable random variables is often sufficient and more easily met.

The assumption is important in the classical form of the central limit theorem, which states that the probability distribution of the sum (or average) of IID variables with finite variance approaches a normal distribution.

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