# F-distribution

*F*-statistics as used in population genetics.

Probability density function | |

Cumulative distribution function | |

Parameters |
d_{1}, d_{2} > 0 deg. of freedom |
---|---|

Support |
x ∈ [0, +∞) |

CDF | |

Mean |
for d_{2} > 2 |

Mode |
for d_{1} > 2 |

Variance |
for d_{2} > 4 |

Skewness |
for d_{2} > 6 |

Ex. kurtosis |
see text |

MGF |
does not exist, raw moments defined in text and in ^{[1]}^{[2]} |

CF |
see text |

In probability theory and statistics, the ** F-distribution**, also known as

**Snedecor's**or the

*F*distribution**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, e.g.,

*F*-test.

^{[1]}

^{[2]}

^{[3]}

^{[4]}

## Definition

If a random variable *X* has an *F*-distribution with parameters *d*_{1} and *d*_{2}, we write *X* ~ F(*d*_{1}, *d*_{2}). Then the probability density function (pdf) for *X* is given by

for real *x* ≥ 0. Here is the beta function. In many applications, the parameters *d*_{1} and *d*_{2} 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(*d*_{1}, *d*_{2}) are given in the sidebox; for *d*_{2} > 8, the excess kurtosis is

- .

The *k*-th moment of an F(*d*_{1}, *d*_{2}) distribution exists and is finite only when 2*k* < *d*_{2} and it is equal to ^{[5]}

The *F*-distribution is a particular parametrization of the beta prime distribution, which is also called the beta distribution of the second kind.

The characteristic function is listed incorrectly in many standard references (e.g.,^{[2]}). The correct expression ^{[6]} is

where *U*(*a*, *b*, *z*) is the confluent hypergeometric function of the second kind.

## Characterization

A random variate of the *F*-distribution with parameters *d*_{1} and *d*_{2} arises as the ratio of two appropriately scaled chi-squared variates:^{[7]}

where

*U*_{1}and*U*_{2}have chi-squared distributions with*d*_{1}and*d*_{2}degrees of freedom respectively, and*U*_{1}and*U*_{2}are independent.

In instances where the *F*-distribution is used, for example in the analysis of variance, independence of *U*_{1} and *U*_{2} might be demonstrated by applying Cochran's theorem.

Equivalently, the random variable of the *F*-distribution may also be written

where *s*_{1}^{2} and *s*_{2}^{2} are the sums of squares *S*_{1}^{2} and *S*_{2}^{2} from two normal processes with variances σ_{1}^{2} and σ_{2}^{2} divided by the corresponding number of χ^{2} degrees of freedom, *d*_{1} and *d*_{2} respectively : and .

In a frequentist context, a scaled *F*-distribution therefore gives the probability *p*(*s*_{1}^{2}/*s*_{2}^{2} | σ_{1}^{2}, σ_{2}^{2}), with the *F*-distribution itself, without any scaling, applying where σ_{1}^{2} is being taken equal to σ_{2}^{2}. This is the context in which the *F*-distribution most generally appears in *F*-tests: where the null hypothesis is that two independent normal variances are equal, and the observed sums of some appropriately selected squares are then examined to see whether their ratio is significantly incompatible with this null hypothesis.

The quantity *X* has the same distribution in Bayesian statistics, if an uninformative rescaling-invariant Jeffreys prior is taken for the prior probabilities of σ_{1}^{2} and σ_{2}^{2}.^{[8]} In this context, a scaled *F*-distribution thus gives the posterior probability *p*(σ_{2}^{2}/σ_{1}^{2}|*s*_{1}^{2}, *s*_{2}^{2}), where now the observed sums *s*_{1}^{2} and *s*_{2}^{2} are what are taken as known.

### Differential equation

The probability density function of the *F*-distribution is a solution of the following differential equation:

## Properties and related distributions

- If and are independent, then
- Parametrization 1: If are independent, then
- Parametrization 2: If are independent, then
- If (Beta distribution) then
- Equivalently, if
*X*~ F(*d*_{1},*d*_{2}), then . - If
*X*~ F(*d*_{1},*d*_{2}) then has the chi-squared distribution - F(
*d*_{1},*d*_{2}) is equivalent to the scaled Hotelling's T-squared distribution . - If
*X*~ F(*d*_{1},*d*_{2}) then*X*^{−1}~ F(*d*_{2},*d*_{1}). - If
*X*~ t(*n*) -- Student's t-distribution -- then:

*F*-distribution is a special case of type 6 Pearson distribution- If
*X*and*Y*are independent, with*X*,*Y*~ Laplace(μ,*b*) then

- If
*X*~ F(*n*,*m*) then (Fisher's z-distribution) - The noncentral
*F*-distribution simplifies to the*F*-distribution if λ = 0. - The doubly noncentral
*F*-distribution simplifies to the*F*-distribution if - If is the quantile
*p*for*X*~ F(*d*_{1},*d*_{2}) and is the quantile 1−*p*for*Y*~ F(*d*_{2},*d*_{1}), then

- .

## See also

## References

- 1 2 Johnson, Norman Lloyd; Samuel Kotz; N. Balakrishnan (1995).
*Continuous Univariate Distributions, Volume 2 (Second Edition, Section 27)*. Wiley. ISBN 0-471-58494-0. - 1 2 3 Abramowitz, Milton; Stegun, Irene Ann, eds. (1983) [June 1964]. "Chapter 26".
*Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables*. Applied Mathematics Series.**55**(Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C., USA; New York, USA: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 946. ISBN 0-486-61272-4. LCCN 64-60036. MR 0167642. ISBN 978-0-486-61272-0. LCCN 65-12253. - ↑ NIST (2006). Engineering Statistics Handbook – F Distribution
- ↑ Mood, Alexander; Franklin A. Graybill; Duane C. Boes (1974).
*Introduction to the Theory of Statistics (Third Edition, pp. 246–249)*. McGraw-Hill. ISBN 0-07-042864-6. - ↑ Taboga, Marco. "The F distribution".
- ↑ Phillips, P. C. B. (1982) "The true characteristic function of the F distribution,"
*Biometrika*, 69: 261–264 JSTOR 2335882 - ↑ M.H. DeGroot (1986),
*Probability and Statistics*(2nd Ed), Addison-Wesley. ISBN 0-201-11366-X, p. 500 - ↑ G.E.P. Box and G.C. Tiao (1973),
*Bayesian Inference in Statistical Analysis*, Addison-Wesley. p.110

## External links

- Table of critical values of the
*F*-distribution - Earliest Uses of Some of the Words of Mathematics: entry on
*F*-distribution contains a brief history - Free calculator for
*F*-testing