In probability theory and statistics, a sequence or other collection of random variables is independent and identically distributed (i.i.d.) if each random variable has the same probability distribution as the others and all are mutually independent.
The abbreviation i.i.d. is particularly common in statistics (often as iid, sometimes written IID), where observations in a sample are often assumed to be effectively i.i.d. for the purposes of statistical inference. The assumption (or requirement) that observations be i.i.d. 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. 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 i.i.d. variables with finite variance approaches a normal distribution.
Note that IID refers to sequences of random variables. "Independent and identically distributed" implies an element in the sequence is independent of the random variables that came before it. In this way, an IID sequence is different from a Markov sequence, where the probability distribution for the nth random variable is a function of the previous random variable in the sequence (for a first order Markov sequence). An IID sequence does not imply the probabilities for all elements of the sample space or event space must be the same. For example, repeated throws of loaded dice will produce a sequence that is IID, despite the outcomes being biased.
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—Paul Valéry (1871–1945)