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In
probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...
and statistics, the ''F''-distribution or F-ratio, also known as Snedecor's ''F'' distribution or the Fisher–Snedecor distribution (after
Ronald Fisher Sir Ronald Aylmer Fisher (17 February 1890 – 29 July 1962) was a British polymath who was active as a mathematician, statistician, biologist, geneticist, and academic. For his work in statistics, he has been described as "a genius who ...
and
George W. Snedecor George Waddel Snedecor (October 20, 1881 – February 15, 1974) was an American mathematician and statistician. He contributed to the foundations of analysis of variance, data analysis, experimental design, and statistical methodology. Snedecor's ...
) is a
continuous probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon ...
that arises frequently as the
null distribution In statistical hypothesis testing, the null distribution is the probability distribution of the test statistic when the null hypothesis is true. For example, in an F-test, the null distribution is an F-distribution. Null distribution is a tool scie ...
of a
test statistic A test statistic is a statistic (a quantity derived from the sample) used in statistical hypothesis testing.Berger, R. L.; Casella, G. (2001). ''Statistical Inference'', Duxbury Press, Second Edition (p.374) A hypothesis test is typically specifie ...
, most notably in the
analysis of variance Analysis of variance (ANOVA) is a collection of statistical models and their associated estimation procedures (such as the "variation" among and between groups) used to analyze the differences among means. ANOVA was developed by the statistician ...
(ANOVA) and other ''F''-tests.


Definition

The F-distribution with ''d''1 and ''d''2 degrees of freedom is the distribution of : X = \frac where S_1 and S_2 are independent
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the p ...
s with
chi-square distribution In probability theory and statistics, the chi-squared distribution (also chi-square or \chi^2-distribution) with k degrees of freedom is the distribution of a sum of the squares of k independent standard normal random variables. The chi-square ...
s with respective degrees of freedom d_1 and d_2. It can be shown to follow that the
probability density function In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) c ...
(pdf) for ''X'' is given by : \begin f(x; d_1,d_2) &= \frac \\ pt&=\frac \left(\frac\right)^ x^ \left(1+\frac \, x \right)^ \end for
real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (201 ...
''x'' > 0. Here \mathrm is the
beta function In mathematics, the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial coefficients. It is defined by the integral : \Beta(z_1,z_2) = \int_0^1 t^ ...
. In many applications, the parameters ''d''1 and ''d''2 are
positive integer In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...
s, but the distribution is well-defined for positive real values of these parameters. The
cumulative distribution function In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Ev ...
is :F(x; d_1,d_2)=I_\left (\tfrac, \tfrac \right) , where ''I'' is the
regularized incomplete beta function In mathematics, the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial coefficients. It is defined by the integral : \Beta(z_1,z_2) = \int_0^1 t^ ...
. 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 :\gamma_2 = 12\frac. 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 :\mu _X(k) =\left( \frac\right)^k \frac \frac.   The ''F''-distribution is a particular parametrization of the
beta prime distribution In probability theory and statistics, the beta prime distribution (also known as inverted beta distribution or beta distribution of the second kindJohnson et al (1995), p 248) is an absolutely continuous probability distribution. Definitions ...
, which is also called the beta distribution of the second kind. The
characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts: * The indicator function of a subset, that is the function ::\mathbf_A\colon X \to \, :which for a given subset ''A'' of ''X'', has value 1 at point ...
is listed incorrectly in many standard references (e.g.,). The correct expression is :\varphi^F_(s) = \frac U \! \left(\frac,1-\frac,-\frac \imath s \right) where ''U''(''a'', ''b'', ''z'') is the
confluent hypergeometric function In mathematics, a confluent hypergeometric function is a solution of a confluent hypergeometric equation, which is a degenerate form of a hypergeometric differential equation where two of the three regular singularities merge into an irregular ...
of the second kind.


Characterization

A
random variate In probability and statistics, a random variate or simply variate is a particular outcome of a ''random variable'': the random variates which are other outcomes of the same random variable might have different values ( random numbers). A random ...
of the ''F''-distribution with parameters d_1 and d_2 arises as the ratio of two appropriately scaled chi-squared variates: :X = \frac where *U_1 and U_2 have
chi-squared distribution In probability theory and statistics, the chi-squared distribution (also chi-square or \chi^2-distribution) with k degrees of freedom is the distribution of a sum of the squares of k independent standard normal random variables. The chi-squar ...
s with d_1 and d_2
degrees of freedom Degrees of freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a thermodynamic system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or ...
respectively, and *U_1 and U_2 are
independent Independent or Independents may refer to: Arts, entertainment, and media Artist groups * Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s * Independe ...
. In instances where the ''F''-distribution is used, for example in the
analysis of variance Analysis of variance (ANOVA) is a collection of statistical models and their associated estimation procedures (such as the "variation" among and between groups) used to analyze the differences among means. ANOVA was developed by the statistician ...
, independence of U_1 and U_2 might be demonstrated by applying
Cochran's theorem In statistics, Cochran's theorem, devised by William G. Cochran, is a theorem used to justify results relating to the probability distributions of statistics that are used in the analysis of variance. Statement Let ''U''1, ..., ''U'N'' be i.i. ...
. Equivalently, the random variable of the ''F''-distribution may also be written :X = \frac \div \frac, where s_1^2 = \frac and s_2^2 = \frac, S_1^2 is the sum of squares of d_1 random variables from normal distribution N(0,\sigma_1^2) and S_2^2 is the sum of squares of d_2 random variables from normal distribution N(0,\sigma_2^2). In a
frequentist Frequentist inference is a type of statistical inference based in frequentist probability, which treats “probability” in equivalent terms to “frequency” and draws conclusions from sample-data by means of emphasizing the frequency or pr ...
context, a scaled ''F''-distribution therefore gives the probability p(s_1^2/s_2^2 \mid \sigma_1^2, \sigma_2^2), with the ''F''-distribution itself, without any scaling, applying where \sigma_1^2 is being taken equal to \sigma_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 In Bayesian probability, the Jeffreys prior, named after Sir Harold Jeffreys, is a non-informative (objective) prior distribution for a parameter space; its density function is proportional to the square root of the determinant of the Fisher infor ...
is taken for the prior probabilities of \sigma_1^2 and \sigma_2^2.G. E. P. Box and G. C. Tiao (1973), ''Bayesian Inference in Statistical Analysis'', Addison-Wesley. p. 110 In this context, a scaled ''F''-distribution thus gives the posterior probability p(\sigma^2_2 /\sigma_1^2 \mid s^2_1, s^2_2), where the observed sums s^2_1 and s^2_2 are now taken as known.


Properties and related distributions

*If X \sim \chi^2_ and Y \sim \chi^2_ (
Chi squared distribution In probability theory and statistics, the chi-squared distribution (also chi-square or \chi^2-distribution) with k degrees of freedom is the distribution of a sum of the squares of k independent standard normal random variables. The chi-square ...
) are
independent Independent or Independents may refer to: Arts, entertainment, and media Artist groups * Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s * Independe ...
, then \frac \sim \mathrm(d_1, d_2) *If X_k \sim \Gamma(\alpha_k,\beta_k)\, (
Gamma distribution In probability theory and statistics, the gamma distribution is a two- parameter family of continuous probability distributions. The exponential distribution, Erlang distribution, and chi-square distribution are special cases of the gamma dis ...
) are independent, then \frac \sim \mathrm(2\alpha_1, 2\alpha_2) *If X \sim \operatorname(d_1/2,d_2/2) (
Beta distribution In probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval , 1in terms of two positive parameters, denoted by ''alpha'' (''α'') and ''beta'' (''β''), that appear as ...
) then \frac \sim \operatorname(d_1,d_2) *Equivalently, if X \sim F(d_1, d_2), then \frac \sim \operatorname(d_1/2,d_2/2). *If X \sim F(d_1, d_2), then \fracX has a
beta prime distribution In probability theory and statistics, the beta prime distribution (also known as inverted beta distribution or beta distribution of the second kindJohnson et al (1995), p 248) is an absolutely continuous probability distribution. Definitions ...
: \fracX \sim \operatorname\left(\tfrac,\tfrac\right). *If X \sim F(d_1, d_2) then Y = \lim_ d_1 X has the
chi-squared distribution In probability theory and statistics, the chi-squared distribution (also chi-square or \chi^2-distribution) with k degrees of freedom is the distribution of a sum of the squares of k independent standard normal random variables. The chi-squar ...
\chi^2_ *F(d_1, d_2) is equivalent to the scaled Hotelling's T-squared distribution \frac \operatorname^2 (d_1, d_1 +d_2-1) . *If X \sim F(d_1, d_2) then X^ \sim F(d_2, d_1). *If X\sim t_Student's t-distribution — then: \begin X^ &\sim \operatorname(1, n) \\ X^ &\sim \operatorname(n, 1) \end *''F''-distribution is a special case of type 6 Pearson distribution *If X and Y are independent, with X,Y\sim Laplace(''μ'', ''b'') then \frac \sim \operatorname(2,2) *If X\sim F(n,m) then \tfrac \sim \operatorname(n,m) (
Fisher's z-distribution Fisher's ''z''-distribution is the statistical distribution of half the logarithm of an ''F''-distribution variate: : z = \frac 1 2 \log F It was first described by Ronald Fisher in a paper delivered at the International Mathematical Congr ...
) *The noncentral ''F''-distribution simplifies to the ''F''-distribution if \lambda=0. *The doubly noncentral ''F''-distribution simplifies to the ''F''-distribution if \lambda_1 = \lambda_2 = 0 *If \operatorname_X(p) is the quantile ''p'' for X\sim F(d_1,d_2) and \operatorname_Y(1-p) is the quantile 1-p for Y\sim F(d_2,d_1), then \operatorname_X(p)=\frac. * ''F''-distribution is an instance of ratio distributions


See also

*
Beta prime distribution In probability theory and statistics, the beta prime distribution (also known as inverted beta distribution or beta distribution of the second kindJohnson et al (1995), p 248) is an absolutely continuous probability distribution. Definitions ...
*
Chi-square distribution In probability theory and statistics, the chi-squared distribution (also chi-square or \chi^2-distribution) with k degrees of freedom is the distribution of a sum of the squares of k independent standard normal random variables. The chi-square ...
*
Chow test The Chow test (), proposed by econometrician Gregory Chow in 1960, is a test of whether the true coefficients in two linear regressions on different data sets are equal. In econometrics, it is most commonly used in time series analysis to test fo ...
*
Gamma distribution In probability theory and statistics, the gamma distribution is a two- parameter family of continuous probability distributions. The exponential distribution, Erlang distribution, and chi-square distribution are special cases of the gamma dis ...
* Hotelling's T-squared distribution *
Wilks' lambda distribution In statistics, Wilks' lambda distribution (named for Samuel S. Wilks), is a probability distribution used in multivariate hypothesis testing, especially with regard to the likelihood-ratio test and multivariate analysis of variance (MANOVA). Defin ...
*
Wishart distribution In statistics, the Wishart distribution is a generalization to multiple dimensions of the gamma distribution. It is named in honor of John Wishart, who first formulated the distribution in 1928. It is a family of probability distributions defin ...


References


External links


Table of critical values of the ''F''-distributionEarliest Uses of Some of the Words of Mathematics: entry on ''F''-distribution contains a brief history
{{DEFAULTSORT:F-distribution Continuous distributions Analysis of variance