Fdistribution
Probability density function


Cumulative distribution function


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

Support  x ∈ [0, +∞) 
\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}\right)}\!  
CDF  I_{\frac{d_1 x}{d_1 x + d_2}} \left(\tfrac{d_1}{2}, \tfrac{d_2}{2} \right) 
Mean 
\frac{d_2}{d_22}\! for d_{2} > 2 
Mode 
\frac{d_12}{d_1}\;\frac{d_2}{d_2+2} for d_{1} > 2 
Variance 
\frac{2\,d_2^2\,(d_1+d_22)}{d_1 (d_22)^2 (d_24)}\! for d_{2} > 4 
Skewness 
\frac{(2 d_1 + d_2  2) \sqrt{8 (d_24)}}{(d_26) \sqrt{d_1 (d_1 + d_2 2)}}\! for d_{2} > 6 
Ex. kurtosis  see text 
MGF  does not exist, raw moments defined in text and in ^{[1]}^{[2]} 
CF  see text 
The Fdistribution, also known as Snedecor's F distribution or the Fisher–Snedecor distribution (after probability theory and statistics, a continuous probability distribution.^{[1]}^{[2]}^{[3]}^{[4]} The Fdistribution arises frequently as the null distribution of a test statistic, most notably in the analysis of variance; see Ftest.
Contents
 Definition 1

Characterization 2
 Differential equation 2.1
 Generalization 3
 Related distributions and properties 4
 See also 5
 References 6
 External links 7
Definition
If a random variable X has an Fdistribution 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
 \begin{align} 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}\right)} \\ &=\frac{1}{\mathrm{B}\!\left(\frac{d_1}{2},\frac{d_2}{2}\right)} \left(\frac{d_1}{d_2}\right)^{\frac{d_1}{2}} x^{\frac{d_1}{2}  1} \left(1+\frac{d_1}{d_2}\,x\right)^{\frac{d_1+d_2}{2}} \end{align}
for real x ≥ 0. Here \mathrm{B} is the beta function. In many applications, the parameters d_{1} and d_{2} are positive integers, but the distribution is welldefined for positive real values of these parameters.
The cumulative distribution function is
 F(x; d_1,d_2)=I_{\frac{d_1 x}{d_1 x + d_2}}\left (\tfrac{d_1}{2}, \tfrac{d_2}{2} \right) ,
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
 \gamma_2 = 12\frac{d_1(5d_222)(d_1+d_22)+(d_24)(d_22)^2}{d_1(d_26)(d_28)(d_1+d_22)}.
The kth moment of an F(d_{1}, d_{2}) distribution exists and is finite only when 2k < d_{2} and it is equal to ^{[5]}
 \mu _{X}(k) =\left( \frac{d_{2}}{d_{1}}\right)^{k}\frac{\Gamma \left(\tfrac{d_1}{2}+k\right) }{\Gamma \left(\tfrac{d_1}{2}\right) }\frac{\Gamma \left(\tfrac{d_2}{2}k\right) }{\Gamma \left( \tfrac{d_2}{2}\right) }
The Fdistribution 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
 \varphi^F_{d_1, d_2}(s) = \frac{\Gamma(\frac{d_1+d_2}{2})}{\Gamma(\tfrac{d_2}{2})} U \! \left(\frac{d_1}{2},1\frac{d_2}{2},\frac{d_2}{d_1} \imath s \right)
where U(a, b, z) is the confluent hypergeometric function of the second kind.
Characterization
A random variate of the Fdistribution with parameters d_{1} and d_{2} arises as the ratio of two appropriately scaled chisquared variates:^{[7]}
 X = \frac{U_1/d_1}{U_2/d_2}
where
 U_{1} and U_{2} have chisquared distributions with d_{1} and d_{2} degrees of freedom respectively, and
 U_{1} and U_{2} are independent.
In instances where the Fdistribution 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 Fdistribution may also be written
 X = \frac{s_1^2}{\sigma_1^2} \;/\; \frac{s_2^2}{\sigma_2^2}
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.
In a frequentist context, a scaled Fdistribution therefore gives the probability p(s_{1}^{2}/s_{2}^{2}  σ_{1}^{2}, σ_{2}^{2}), with the Fdistribution itself, without any scaling, applying where σ_{1}^{2} is being taken equal to σ_{2}^{2}. This is the context in which the Fdistribution most generally appears in Ftests: 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 rescalinginvariant Jeffreys prior is taken for the prior probabilities of σ_{1}^{2} and σ_{2}^{2}.^{[8]} In this context, a scaled Fdistribution 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 Fdistribution is a solution of the following differential equation:
 \left\{\begin{array}{l} 2 x \left(d_1 x+d_2\right) f'(x)+\left(2 d_1 x+d_2 d_1 xd_2 d_1+2 d_2\right) f(x)=0, \\[12pt] f(1)=\frac{d_1^{\frac{d_1}{2}} d_2^{\frac{d_2}{2}} \left(d_1+d_2\right){}^{\frac{1}{2} \left(d_1d_2\right)}}{B\left(\frac{d_1}{2},\frac{d_2}{2}\right)} \end{array}\right\}
Generalization
A generalization of the (central) Fdistribution is the noncentral Fdistribution.
Related distributions and properties
 If X \sim \chi^2_{d_1} and Y \sim \chi^2_{d_2} are independent, then \frac{X / d_1}{Y / d_2} \sim \mathrm{F}(d_1, d_2)
 If X \sim \operatorname{Beta}(d_1/2,d_2/2) (Beta distribution) then \frac{d_2 X}{d_1(1X)} \sim \operatorname{F}(d_1,d_2)
 Equivalently, if X ~ F(d_{1}, d_{2}), then \frac{d_1 X/d_2}{1+d_1 X/d_2} \sim \operatorname{Beta}(d_1/2,d_2/2).
 If X ~ F(d_{1}, d_{2}) then Y = \lim_{d_2 \to \infty} d_1 X has the chisquared distribution \chi^2_{d_1}
 F(d_{1}, d_{2}) is equivalent to the scaled Hotelling's Tsquared distribution \frac{d_2}{d_1(d_1+d_21)} \operatorname{T}^2 (d_1, d_1 +d_21) .
 If X ~ F(d_{1}, d_{2}) then X^{−1} ~ F(d_{2}, d_{1}).
 If X ~ t(n) then

 X^{2} \sim \operatorname{F}(1, n)
 X^{2} \sim \operatorname{F}(n, 1)
 Fdistribution is a special case of type 6 Pearson distribution
 If X and Y are independent, with X, Y ~ Laplace(μ, b) then

 \tfrac{X\mu}{Y\mu} \sim \operatorname{F}(2,2)
 If X ~ F(n, m) then \tfrac{\log{X}}{2} \sim \operatorname{FisherZ}(n,m) (Fisher's zdistribution)
 The noncentral Fdistribution simplifies to the Fdistribution if λ = 0.
 The doubly noncentral Fdistribution simplifies to the Fdistribution if \lambda_1 = \lambda_2 = 0
 If \operatorname{Q}_X(p) is the quantile p for X ~ F(d_{1}, d_{2}) and \operatorname{Q}_Y(1p) is the quantile 1−p for Y ~ F(d_{2}, d_{1}), then

 \operatorname{Q}_X(p)=\frac{1}{\operatorname{Q}_Y(1p)}.
See also
References
 ^ ^{a} ^{b} Johnson, Norman Lloyd; Samuel Kotz; N. Balakrishnan (1995). Continuous Univariate Distributions, Volume 2 (Second Edition, Section 27). Wiley.
 ^ ^{a} ^{b} ^{c} Abramowitz, Milton; Stegun, Irene A., eds. (1965), "Chapter 26", .
 ^ 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). McGrawHill.
 ^ 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), AddisonWesley. ISBN 020111366X, p. 500
 ^ G.E.P. Box and G.C. Tiao (1973), Bayesian Inference in Statistical Analysis, AddisonWesley. p.110
External links
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