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 acronym i.i.d. is particularly common in statistics, where observations in a sample are typically 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.

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.

• 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.