The acronymi.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.
In probability theory, a sequence or other collection of random variables is independent and identicallydistributed (i.i.d.) if each has the same probability distribution as the others and all are mutually independent.
is particularly common in statistics (sometimes written IID), where observations in a sample are often assumed to be (more-or-less) i.i.d.
All other things being equal, a sequence of outcomes of spins of a roulette wheel is i.i.d.
The condition of exchangeability is stronger than the assumption of identicaldistribution of the individual random variables in the sequence, and weaker than the assumption that they are independent and identicallydistributed.
De Finetti's theorem states that the probability distribution of any infinite exchangeable sequence of Bernoulli random variables is a "mixture" of the probability distributions of independent and identicallydistributed sequences of Bernoulli random variables.
The independence asserted here is conditional independence, i.e., the Bernoulli random variables in the sequence are conditionally independent given the event that p = 2/3, and are conditionally independent given the event that p = 9/10.
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