F-distribution

In probability theory and statistics, the F-distribution, also known as Snedecor's F distribution or the Fisher–Snedecor distribution (after Ronald Fisher and George W. Snedecor) is a continuous probability distribution that arises frequently as the null distribution of a test statistic, most notably in the analysis of variance (ANOVA), e.g., F-test. In probability theory and statistics, the F-distribution, also known as Snedecor's F distribution or the Fisher–Snedecor distribution (after Ronald Fisher and George W. Snedecor) is a continuous probability distribution that arises frequently as the null distribution of a test statistic, most notably in the analysis of variance (ANOVA), e.g., F-test. If a random variable X has an F-distribution with parameters d1 and d2, we write X ~ F(d1, d2). Then the probability density function (pdf) for X is given by f ( x ; d 1 , d 2 ) = ( d 1 x ) d 1 d 2 d 2 ( d 1 x + d 2 ) d 1 + d 2 x B ( d 1 2 , d 2 2 ) = 1 B ( d 1 2 , d 2 2 ) ( d 1 d 2 ) d 1 2 x d 1 2 − 1 ( 1 + d 1 d 2 x ) − d 1 + d 2 2 {displaystyle {egin{aligned}f(x;d_{1},d_{2})&={frac {sqrt {frac {(d_{1},x)^{d_{1}},,d_{2}^{d_{2}}}{(d_{1},x+d_{2})^{d_{1}+d_{2}}}}}{x,mathrm {B} !left({frac {d_{1}}{2}},{frac {d_{2}}{2}} ight)}}\&={frac {1}{mathrm {B} !left({frac {d_{1}}{2}},{frac {d_{2}}{2}} ight)}}left({frac {d_{1}}{d_{2}}} ight)^{frac {d_{1}}{2}}x^{{frac {d_{1}}{2}}-1}left(1+{frac {d_{1}}{d_{2}}},x ight)^{-{frac {d_{1}+d_{2}}{2}}}end{aligned}}} for real x > 0. Here B {displaystyle mathrm {B} } is the beta function. In many applications, the parameters d1 and d2 are positive integers, but the distribution is well-defined for positive real values of these parameters. The cumulative distribution function is where I is the regularized incomplete beta function. The expectation, variance, and other details about the F(d1, d2) are given in the sidebox; for d2 > 8, the excess kurtosis is The k-th moment of an F(d1, d2) distribution exists and is finite only when 2k < d2 and it is equal to The F-distribution is a particular parametrization of the beta prime distribution, which is also called the beta distribution of the second kind.