INDEPENDENT AND IDENTICALLY-DISTRIBUTED RANDOM VARIABLES

In probability theory, a sequence or other collection of random variables is 'independent and identically distributed (i.i.d.)' if each has the same probability distribution as the others and all are mutually independent.
The abbreviation ''i.i.d.'' is particularly common in statistics (sometimes written ''IID''), where observations in a sample are often assumed to be (more-or-less) 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. But in practical applications this may or may not be realistic.
This is important in the classical form of the central limit theorem, the sum of i.i.d. variables with finite variance approaches a normal distribution.

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Examples

See also

Examples

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

★ All other things being equal, a sequence of outcomes of spins of a roulette wheel is i.i.d. From a practical point of view, an important implication of this is that if the roulette ball lands on 'red', for example, 20 times in a row, the next spin is no more or less likely to be 'black' than on any other spin.

★ All other things being equal, a sequence of die rolls is i.i.d.

★ All other things being equal, a sequence of coin flips is i.i.d.

★ An unbiased random walk is i.i.d. because one step has no effect on the following one.

★ One of the simplest statistical tests, the ''z''-test, is used to test hypotheses about means of random variables. When using the ''z''-test, one assumes (requires) that all observations are i.i.d. in order to satisfy the conditions of the central limit theorem.

See also

★ De Finetti's theorem