Independent and identically distributed random variables

In probability theory and statistics, a collection of random variables is independent and identically distributed if each random variable has the same probability distribution as the others and all are mutually independent. This property is usually abbreviated as i.i.d., iid, or IID. IID was first defined in statistics and finds application in different fields such as data mining and signal processing. Statistics commonly deals with random samples. A random sample can be thought of as a set of objects that are chosen randomly. More formally, it is "a sequence of independent, identically distributed (IID) random data points". In other words, the terms random sample and IID are basically one and the same. In statistics, "random sample" is the typical terminology, but in probability it is more common to say "IID". Identically distributed means that there are no overall trends–the distribution doesn't fluctuate and all items in the sample are taken from the same probability distribution. Independent means that the sample items are all independent events. In other words, they are not connected to each other in any way; knowledge of the value of one variable gives no information about the value of the other and vice versa. It is not necessary for IID variables to be uniformly distributed. Being IID merely requires that they all have the same distribution as each other, and are chosen independently from that distribution, regardless of how uniform or non-uniform their distribution may be. Independent and identically distributed random variables are often used as an assumption, which tends to simplify the underlying mathematics. In practical applications of statistical modeling, however, the assumption may or may not be realistic. The i.i.d. assumption is also used in 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. Often the i.i.d. assumption arises in the context of sequences of random variables.

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