Independent and identically distributed random variables

"IID" and "iid" redirect here. For other uses, see IID (disambiguation).

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.[1]

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. To test how realistic the assumption is on a given data set, the autocorrelation can be computed, lag plots drawn or turning point test performed.[2] 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.[3] For example, repeated throws of loaded dice will produce a sequence that is IID, despite the outcomes being biased.

Examples

Uses in modeling

The following are examples or applications of independent and identically distributed (i.i.d.) random variables:

Uses in inference

Generalizations

Many results that were first proven under the assumption that the random variables are i.i.d. have been shown to be true even under a weaker distributional assumption.

Exchangeable random variables

The most general notion which shares the main properties of i.i.d. variables are exchangeable random variables, introduced by Bruno de Finetti. Exchangeability means that while variables may not be independent, future ones behave like past ones – formally, any value of a finite sequence is as likely as any permutation of those values – the joint probability distribution is invariant under the symmetric group.

This provides a useful generalization – for example, sampling without replacement is not independent, but is exchangeable – and is widely used in Bayesian statistics.

Lévy process

Main article: Lévy process

In stochastic calculus, i.i.d. variables are thought of as a discrete time Lévy process: each variable gives how much one changes from one time to another. For example, a sequence of Bernoulli trials is interpreted as the Bernoulli process. One may generalize this to include continuous time Lévy processes, and many Lévy processes can be seen as limits of i.i.d. variables—for instance, the Wiener process is the limit of the Bernoulli process.

White noise

White noise is a simple example of IID.

See also

References

  1. Aaron Clauset. "A brief primer on probability distributions" (PDF). Santa Fe Institute.
  2. Le Boudec, Jean-Yves (2010). Performance Evaluation Of Computer And Communication Systems (PDF). EPFL Press. pp. 46–47. ISBN 978-2-940222-40-7.
  3. Cover, Thomas (2006). Elements Of Information Theory. Wiley-Interscience. pp. 57–58. ISBN 978-0-471-24195-9.
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