## Quick Reference

If *Y*_{1} and *Y*_{2} are independent chi-squared random variables with *ν*_{1} and *ν*_{2} degrees of freedom, respectively, then the ratio *X*, given by , is said to have an *F*-distribution with *ν*_{1} and *ν*_{2} degrees of freedom. This may be written as the *F** _{ν1,ν2}*-distribution. Evidently 1/

*X*will have a

*F*

*-distribution.*

_{ν2,ν1}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 *F** _{ν1, ν2}*-distribution is given by , where B is the beta function. The distribution has mean

*ν*

_{2}/(

*ν*

_{2}−2) provided that

*ν*

_{2}>2. The distribution has variance , provided that

*ν*

_{2}>4. If

*ν*

_{1}≤2 there is a mode at 0, otherwise the mode is at . If

*X*has a

*t*-distribution with

*ν*degrees of freedom, then

*X*

^{2}has an

*F*

_{1,ν}distribution.

*F*-**distribution.** All *F*-distributions take values from 0 to ∞ and, for *ν*_{1}, *ν*_{2} > 4, have mean and mode near 1.

*Subjects:*
Probability and Statistics.

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