- Independent and identically-distributed random variables
:"IID" or "iid" redirects here. For other uses, see:"

IID (disambiguation) .In

probability theory , asequence or other collection ofrandom variable s is**independent and identically distributed (i.i.d.)**if each has the sameprobability distribution as the others and all are mutually independent.The

abbreviation "i.i.d." is particularly common instatistics (often as "iid", sometimes written "IID"), where observations in a sample are often assumed to be (more-or-less) i.i.d. for the purposes ofstatistical inference . The assumption (or requirement) that observations be i.i.d. tends to simplify the underlying mathematics of many statistical methods. However, in practical applications this may or may not be realistic.This 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 finitevariance approaches anormal distribution .**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 dice rolls is i.i.d.

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

*One of the simplest statistical tests, the "z"-test, is used to test hypotheses about

mean s 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 thecentral limit theorem .*In

signal processing andimage processing the notion of transformation to IID implies two specifications, the "ID" part and the "I" part: (ID) the signal level must be balanced on the time axis, and (I=Independent) the signal spectrum must be flattened, i.e. transformed by filtering (such asdeconvolution ) to a white signal (one where all frequencies are equally present).**ee also***

De Finetti's theorem

*Chebyshev's theorem

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2010.*