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.
Famous quotes containing the words independent, random and/or variables:
“There are two kinds of timidity—timidity of mind, and timidity of the nerves; physical timidity, and moral timidity. Each is independent of the other. The body may be frightened and quake while the mind remains calm and bold, and vice versë. This is the key to many eccentricities of conduct. When both kinds meet in the same man he will be good for nothing all his life.”
—Honoré De Balzac (1799–1850)
“... the random talk of people who have no chance of immortality and thus can speak their minds out has a setting, often, of lights, streets, houses, human beings, beautiful or grotesque, which will weave itself into the moment for ever.”
—Virginia Woolf (1882–1941)
“The variables are surprisingly few.... One can whip or be whipped; one can eat excrement or quaff urine; mouth and private part can be meet in this or that commerce. After which there is the gray of morning and the sour knowledge that things have remained fairly generally the same since man first met goat and woman.”
—George Steiner (b. 1929)