Encyclopedia > Independent 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 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. |