Continuous probability distribution
Not to be confused with
F -statistics as used in population genetics.
Fisher–Snedecor Probability density function
Cumulative distribution function
Parameters d 1 ,d 2 > 0 deg. of freedomSupport x ∈ ( 0 , + ∞ ) {\displaystyle x\in (0,+\infty )\;} d 1 = 1 {\displaystyle d_{1}=1} x ∈ [ 0 , + ∞ ) {\displaystyle x\in [0,+\infty )\;} PDF ( 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 ) {\displaystyle {\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 d 1 x d 1 x + d 2 ( d 1 2 , d 2 2 ) {\displaystyle I_{\frac {d_{1}x}{d_{1}x+d_{2}}}\left({\tfrac {d_{1}}{2}},{\tfrac {d_{2}}{2}}\right)} Mean d 2 d 2 − 2 {\displaystyle {\frac {d_{2}}{d_{2}-2}}\!} d 2 > 2Mode d 1 − 2 d 1 d 2 d 2 + 2 {\displaystyle {\frac {d_{1}-2}{d_{1}}}\;{\frac {d_{2}}{d_{2}+2}}} d 1 > 2Variance 2 d 2 2 ( d 1 + d 2 − 2 ) d 1 ( d 2 − 2 ) 2 ( d 2 − 4 ) {\displaystyle {\frac {2\,d_{2}^{2}\,(d_{1}+d_{2}-2)}{d_{1}(d_{2}-2)^{2}(d_{2}-4)}}\!} d 2 > 4Skewness ( 2 d 1 + d 2 − 2 ) 8 ( d 2 − 4 ) ( d 2 − 6 ) d 1 ( d 1 + d 2 − 2 ) {\displaystyle {\frac {(2d_{1}+d_{2}-2){\sqrt {8(d_{2}-4)}}}{(d_{2}-6){\sqrt {d_{1}(d_{1}+d_{2}-2)}}}}\!} d 2 > 6Excess kurtosis see text Entropy ln Γ ( d 1 2 ) + ln Γ ( d 2 2 ) − ln Γ ( d 1 + d 2 2 ) + ( 1 − d 1 2 ) ψ ( 1 + d 1 2 ) − ( 1 + d 2 2 ) ψ ( 1 + d 2 2 ) + ( d 1 + d 2 2 ) ψ ( d 1 + d 2 2 ) + ln d 2 d 1 {\displaystyle {\begin{aligned}&\ln \Gamma {\left({\tfrac {d_{1}}{2}}\right)}+\ln \Gamma {\left({\tfrac {d_{2}}{2}}\right)}-\ln \Gamma {\left({\tfrac {d_{1}+d_{2}}{2}}\right)}\\&+\left(1-{\tfrac {d_{1}}{2}}\right)\psi {\left(1+{\tfrac {d_{1}}{2}}\right)}-\left(1+{\tfrac {d_{2}}{2}}\right)\psi {\left(1+{\tfrac {d_{2}}{2}}\right)}\\&+\left({\tfrac {d_{1}+d_{2}}{2}}\right)\psi {\left({\tfrac {d_{1}+d_{2}}{2}}\right)}+\ln {\frac {d_{2}}{d_{1}}}\end{aligned}}} [ 1] MGF does not exist, raw moments defined in text and in[ 2] [ 3] CF see text
Inprobability theory andstatistics , theF -distributionF -ratioSnedecor'sF distribution or theFisher–Snedecor distribution (afterRonald Fisher andGeorge W. Snedecor ), is acontinuous probability distribution that arises frequently as thenull distribution of atest statistic , most notably in theanalysis of variance (ANOVA) and otherF -tests[ 2] [ 3] [ 4] [ 5]
TheF -distribution withd 1 andd 2 degrees of freedom is the distribution of
X = U 1 / d 1 U 2 / d 2 {\displaystyle X={\frac {U_{1}/d_{1}}{U_{2}/d_{2}}}}
whereU 1 {\textstyle U_{1}} U 2 {\textstyle U_{2}} independent random variables withchi-square distributions with respective degrees of freedomd 1 {\textstyle d_{1}} d 2 {\textstyle d_{2}}
It can be shown to follow that theprobability density function (pdf) forX 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 {\begin{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\operatorname {B} \left({\frac {d_{1}}{2}},{\frac {d_{2}}{2}}\right)}}\\[5pt]&={\frac {1}{\operatorname {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{aligned}}}
forreal x > 0. HereB {\displaystyle \mathrm {B} } beta function . In many applications, the parametersd 1 andd 2 arepositive integers , but the distribution is well-defined for positive real values of these parameters.
Thecumulative distribution function is
F ( x ; d 1 , d 2 ) = I d 1 x / ( d 1 x + d 2 ) ( d 1 2 , d 2 2 ) , {\displaystyle F(x;d_{1},d_{2})=I_{d_{1}x/(d_{1}x+d_{2})}\left({\tfrac {d_{1}}{2}},{\tfrac {d_{2}}{2}}\right),}
whereI is theregularized incomplete beta function .
The expectation, variance, and other details about the F(d 1 ,d 2 ) are given in the sidebox; ford 2 > 8, theexcess kurtosis is
γ 2 = 12 d 1 ( 5 d 2 − 22 ) ( d 1 + d 2 − 2 ) + ( d 2 − 4 ) ( d 2 − 2 ) 2 d 1 ( d 2 − 6 ) ( d 2 − 8 ) ( d 1 + d 2 − 2 ) . {\displaystyle \gamma _{2}=12{\frac {d_{1}(5d_{2}-22)(d_{1}+d_{2}-2)+(d_{2}-4)(d_{2}-2)^{2}}{d_{1}(d_{2}-6)(d_{2}-8)(d_{1}+d_{2}-2)}}.}
Thek -th moment of an F(d 1 ,d 2 ) distribution exists and is finite only when 2k <d 2 and it is equal to[ 6]
μ X ( k ) = ( d 2 d 1 ) k Γ ( d 1 2 + k ) Γ ( d 1 2 ) Γ ( d 2 2 − k ) Γ ( d 2 2 ) . {\displaystyle \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)}}.}
TheF -distribution is a particularparametrization of thebeta prime distribution , which is also called the beta distribution of the second kind.
Thecharacteristic function is listed incorrectly in many standard references (e.g.,[ 3] [ 7]
φ d 1 , d 2 F ( s ) = Γ ( d 1 + d 2 2 ) Γ ( d 2 2 ) U ( d 1 2 , 1 − d 2 2 , − d 2 d 1 ı s ) {\displaystyle \varphi _{d_{1},d_{2}}^{F}(s)={\frac {\Gamma {\left({\frac {d_{1}+d_{2}}{2}}\right)}}{\Gamma {\left({\tfrac {d_{2}}{2}}\right)}}}U\!\left({\frac {d_{1}}{2}},1-{\frac {d_{2}}{2}},-{\frac {d_{2}}{d_{1}}}\imath s\right)}
whereU (a ,b ,z ) is theconfluent hypergeometric function of the second kind.
Related distributions [ edit ] Relation to the chi-squared distribution [ edit ] In instances where theF -distribution is used, for example in theanalysis of variance , independence ofU 1 {\displaystyle U_{1}} U 2 {\displaystyle U_{2}} Cochran's theorem .
Equivalently, since thechi-squared distribution is the sum of squares ofindependent standard normal random variables, the random variable of theF -distribution may also be written
X = s 1 2 σ 1 2 ÷ s 2 2 σ 2 2 , {\displaystyle X={\frac {s_{1}^{2}}{\sigma _{1}^{2}}}\div {\frac {s_{2}^{2}}{\sigma _{2}^{2}}},}
wheres 1 2 = S 1 2 d 1 {\displaystyle s_{1}^{2}={\frac {S_{1}^{2}}{d_{1}}}} s 2 2 = S 2 2 d 2 {\displaystyle s_{2}^{2}={\frac {S_{2}^{2}}{d_{2}}}} S 1 2 {\displaystyle S_{1}^{2}} d 1 {\displaystyle d_{1}} N ( 0 , σ 1 2 ) {\displaystyle N(0,\sigma _{1}^{2})} S 2 2 {\displaystyle S_{2}^{2}} d 2 {\displaystyle d_{2}} N ( 0 , σ 2 2 ) {\displaystyle N(0,\sigma _{2}^{2})}
In afrequentist context, a scaledF -distribution therefore gives the probabilityp ( s 1 2 / s 2 2 ∣ σ 1 2 , σ 2 2 ) {\displaystyle p(s_{1}^{2}/s_{2}^{2}\mid \sigma _{1}^{2},\sigma _{2}^{2})} F -distribution itself, without any scaling, applying whereσ 1 2 {\displaystyle \sigma _{1}^{2}} σ 2 2 {\displaystyle \sigma _{2}^{2}} F -distribution most generally appears inF -tests
The quantityX {\displaystyle X} Jeffreys prior is taken for theprior probabilities ofσ 1 2 {\displaystyle \sigma _{1}^{2}} σ 2 2 {\displaystyle \sigma _{2}^{2}} [ 8] F -distribution thus gives the posterior probabilityp ( σ 2 2 / σ 1 2 ∣ s 1 2 , s 2 2 ) {\displaystyle p(\sigma _{2}^{2}/\sigma _{1}^{2}\mid s_{1}^{2},s_{2}^{2})} s 1 2 {\displaystyle s_{1}^{2}} s 2 2 {\displaystyle s_{2}^{2}}
IfX ∼ χ d 1 2 {\displaystyle X\sim \chi _{d_{1}}^{2}} Y ∼ χ d 2 2 {\displaystyle Y\sim \chi _{d_{2}}^{2}} Chi squared distribution ) areindependent , thenX / d 1 Y / d 2 ∼ F ( d 1 , d 2 ) {\displaystyle {\frac {X/d_{1}}{Y/d_{2}}}\sim \mathrm {F} (d_{1},d_{2})} IfX k ∼ Γ ( α k , β k ) {\displaystyle X_{k}\sim \Gamma (\alpha _{k},\beta _{k})\,} Gamma distribution ) are independent, thenα 2 β 1 X 1 α 1 β 2 X 2 ∼ F ( 2 α 1 , 2 α 2 ) {\displaystyle {\frac {\alpha _{2}\beta _{1}X_{1}}{\alpha _{1}\beta _{2}X_{2}}}\sim \mathrm {F} (2\alpha _{1},2\alpha _{2})} IfX ∼ Beta ( d 1 / 2 , d 2 / 2 ) {\displaystyle X\sim \operatorname {Beta} (d_{1}/2,d_{2}/2)} Beta distribution ) thend 2 X d 1 ( 1 − X ) ∼ F ( d 1 , d 2 ) {\displaystyle {\frac {d_{2}X}{d_{1}(1-X)}}\sim \operatorname {F} (d_{1},d_{2})} Equivalently, ifX ∼ F ( d 1 , d 2 ) {\displaystyle X\sim F(d_{1},d_{2})} d 1 X / d 2 1 + d 1 X / d 2 ∼ Beta ( d 1 / 2 , d 2 / 2 ) {\displaystyle {\frac {d_{1}X/d_{2}}{1+d_{1}X/d_{2}}}\sim \operatorname {Beta} (d_{1}/2,d_{2}/2)} IfX ∼ F ( d 1 , d 2 ) {\displaystyle X\sim F(d_{1},d_{2})} d 1 d 2 X {\displaystyle {\frac {d_{1}}{d_{2}}}X} beta prime distribution :d 1 d 2 X ∼ β ′ ( d 1 2 , d 2 2 ) {\displaystyle {\frac {d_{1}}{d_{2}}}X\sim \operatorname {\beta ^{\prime }} \left({\tfrac {d_{1}}{2}},{\tfrac {d_{2}}{2}}\right)} IfX ∼ F ( d 1 , d 2 ) {\displaystyle X\sim F(d_{1},d_{2})} Y = lim d 2 → ∞ d 1 X {\displaystyle Y=\lim _{d_{2}\to \infty }d_{1}X} chi-squared distribution χ d 1 2 {\displaystyle \chi _{d_{1}}^{2}} F ( d 1 , d 2 ) {\displaystyle F(d_{1},d_{2})} Hotelling's T-squared distribution d 2 d 1 ( d 1 + d 2 − 1 ) T 2 ( d 1 , d 1 + d 2 − 1 ) {\displaystyle {\frac {d_{2}}{d_{1}(d_{1}+d_{2}-1)}}\operatorname {T} ^{2}(d_{1},d_{1}+d_{2}-1)} IfX ∼ F ( d 1 , d 2 ) {\displaystyle X\sim F(d_{1},d_{2})} X − 1 ∼ F ( d 2 , d 1 ) {\displaystyle X^{-1}\sim F(d_{2},d_{1})} IfX ∼ t ( n ) {\displaystyle X\sim t_{(n)}} Student's t-distribution — then:X 2 ∼ F ( 1 , n ) X − 2 ∼ F ( n , 1 ) {\displaystyle {\begin{aligned}X^{2}&\sim \operatorname {F} (1,n)\\X^{-2}&\sim \operatorname {F} (n,1)\end{aligned}}} F -distribution is a special case of type 6Pearson distribution IfX {\displaystyle X} Y {\displaystyle Y} X , Y ∼ {\displaystyle X,Y\sim } Laplace(μ ,b ) then| X − μ | | Y − μ | ∼ F ( 2 , 2 ) {\displaystyle {\frac {|X-\mu |}{|Y-\mu |}}\sim \operatorname {F} (2,2)} IfX ∼ F ( n , m ) {\displaystyle X\sim F(n,m)} log X 2 ∼ FisherZ ( n , m ) {\displaystyle {\tfrac {\log {X}}{2}}\sim \operatorname {FisherZ} (n,m)} Fisher's z-distribution ) ThenoncentralF -distribution simplifies to theF -distribution ifλ = 0 {\displaystyle \lambda =0} The doublynoncentralF -distribution simplifies to theF -distribution ifλ 1 = λ 2 = 0 {\displaystyle \lambda _{1}=\lambda _{2}=0} IfQ X ( p ) {\displaystyle \operatorname {Q} _{X}(p)} p forX ∼ F ( d 1 , d 2 ) {\displaystyle X\sim F(d_{1},d_{2})} Q Y ( 1 − p ) {\displaystyle \operatorname {Q} _{Y}(1-p)} 1 − p {\displaystyle 1-p} Y ∼ F ( d 2 , d 1 ) {\displaystyle Y\sim F(d_{2},d_{1})} Q X ( p ) = 1 Q Y ( 1 − p ) . {\displaystyle \operatorname {Q} _{X}(p)={\frac {1}{\operatorname {Q} _{Y}(1-p)}}.} F -distribution is an instance ofratio distributions W -distribution[ 9] Beta prime distribution Chi-square distribution Chow test Gamma distribution Hotelling's T-squared distribution Wilks' lambda distribution Wishart distribution Modified half-normal distribution [ 10] ( 0 , ∞ ) {\displaystyle (0,\infty )} f ( x ) = 2 β α 2 x α − 1 exp ( − β x 2 + γ x ) Ψ ( α 2 , γ β ) {\displaystyle f(x)={\frac {2\beta ^{\frac {\alpha }{2}}x^{\alpha -1}\exp(-\beta x^{2}+\gamma x)}{\Psi {\left({\frac {\alpha }{2}},{\frac {\gamma }{\sqrt {\beta }}}\right)}}}} Ψ ( α , z ) = 1 Ψ 1 ( ( α , 1 2 ) ( 1 , 0 ) ; z ) {\displaystyle \Psi (\alpha ,z)={}_{1}\Psi _{1}\left({\begin{matrix}\left(\alpha ,{\frac {1}{2}}\right)\\(1,0)\end{matrix}};z\right)} Fox–Wright Psi function .^ Lazo, A.V.; Rathie, P. (1978). "On the entropy of continuous probability distributions".IEEE Transactions on Information Theory .24 (1). IEEE:120– 122.doi :10.1109/tit.1978.1055832 . ^a b Johnson, Norman Lloyd; Samuel Kotz; N. Balakrishnan (1995).Continuous Univariate Distributions, Volume 2 (Section 27) (2nd ed.). Wiley.ISBN 0-471-58494-0 ^a b c Abramowitz, Milton ;Stegun, Irene Ann , eds. (1983) [June 1964]."Chapter 26" .Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables ISBN 978-0-486-61272-0 LCCN 64-60036 .MR 0167642 .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 ed.). McGraw-Hill. pp. 246– 249.ISBN 0-07-042864-6 ^ Taboga, Marco."The F distribution" . ^ Phillips, P. C. B. (1982) "The true characteristic function of the F distribution,"Biometrika JSTOR 2335882 ^ Box, G. E. P.; Tiao, G. C. (1973).Bayesian Inference in Statistical Analysis . Addison-Wesley. p. 110.ISBN 0-201-00622-7 ^ Mahmoudi, Amin; Javed, Saad Ahmed (October 2022)."Probabilistic Approach to Multi-Stage Supplier Evaluation: Confidence Level Measurement in Ordinal Priority Approach" .Group Decision and Negotiation .31 (5):1051– 1096.doi :10.1007/s10726-022-09790-1 .ISSN 0926-2644 .PMC 9409630 PMID 36042813 . ^ Sun, Jingchao; Kong, Maiying; Pal, Subhadip (22 June 2021)."The Modified-Half-Normal distribution: Properties and an efficient sampling scheme" (PDF) .Communications in Statistics - Theory and Methods .52 (5):1591– 1613.doi :10.1080/03610926.2021.1934700 .ISSN 0361-0926 .S2CID 237919587 .
Discrete
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Continuous
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Mixed
Multivariate Directional Degenerate singular Families
