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If Y1 and Y2 are independent chi-squared random variables with ν1 and ν2 degrees of freedom, respectively, then the ratio X, given by , is said to have an F-distribution with ν1 and ν2 degrees of freedom. This may be written as the Fν1,ν2-distribution. Evidently 1/X will have a Fν2,ν1-distribution.

The form of the distribution was first given in 1922 by Sir Ronald Fisher, and it is sometimes still referred to as Fisher's F-distribution. In 1934 the distribution was tabulated (see Percentage Points for the F-Distribution) by Snedecor, who used the letter F in Fisher's honour. The distribution is therefore also referred to as the Snedecor F-distribution. The probability density function f of the Fν1, ν2-distribution is given by , where B is the beta function. The distribution has mean ν2/(ν2−2) provided that ν2>2. The distribution has variance , provided that ν2>4. If ν1≤2 there is a mode at 0, otherwise the mode is at . If X has a t-distribution with ν degrees of freedom, then X2 has an F1,ν distribution.

F-distribution. All F-distributions take values from 0 to ∞ and, for ν1, ν2 > 4, have mean and mode near 1.

Subjects: Probability and Statistics.

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