F-distribution

The F-distribution with d₁ and d₂ degrees of freedom is the distribution of

$$X\; =\; \backslash frac\{U\_1/d\_1\}\{U\_2/d\_2\}$$

where $U\_1$ and $U\_2$ are independent random variables with chi-square distributions with respective degrees of freedom $d\_1$ and $d\_2$.

It can be shown to follow that the probability density function (pdf) for X is given by

$$\backslash begin\{align\}$$

f(x; d_1,d_2) &= \frac{\sqrt{\frac{(d_1x)^{d_1}\,\,d_2^{d_2}} {(d_1x+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{align}

for real x > 0. Here $\backslash mathrm\{B\}$ 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

$$F(x;\; d\_1,d\_2)=I\_\{d\_1\; x/(d\_1\; x\; +\; d\_2)\}\backslash left\; (\backslash tfrac\{d\_1\}\{2\},\; \backslash tfrac\{d\_2\}\{2\}\; \backslash right)\; ,$$

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

$$\backslash gamma\_2\; =\; 12\backslash frac\{d\_1(5d\_2-22)(d\_1+d\_2-2)+(d\_2-4)(d\_2-2)^2\}\{d\_1(d\_2-6)(d\_2-8)(d\_1+d\_2-2)\}.$$

The k-th moment of an F(d₁, d₂) distribution exists and is finite only when 2k < d₂ and it is equal to{{cite web | last1 = Taboga | first1 = Marco | url = http://www.statlect.com/F_distribution.htm | title = The F distribution}}

$$\backslash mu\; \_X(k)\; =\backslash left(\; \backslash frac\{d\_2\}\{d\_1\}\backslash right)^k\; \backslash frac\{\backslash Gamma\; \backslash left(\backslash tfrac\{d\_1\}\{2\}+k\backslash right)\; \}\{\backslash Gamma\; \backslash left(\backslash tfrac\{d\_1\}\{2\}\backslash right)\}\; \backslash frac\{\backslash Gamma\; \backslash left(\backslash tfrac\{d\_2\}\{2\}-k\backslash right)\; \}\{\backslash Gamma\; \backslash left(\; \backslash tfrac\{d\_2\}\{2\}\backslash right)\; \}.$$

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.,). The correct expression Phillips, P. C. B. (1982) "The true characteristic function of the F distribution," Biometrika, 69: 261–264 {{JSTOR|2335882}} is

$$\backslash varphi^F\_\{d\_1,\; d\_2\}(s)\; =\; \backslash frac\{\backslash Gamma\{\backslash left(\backslash frac\{d\_1+d\_2\}\{2\}\backslash right)\}\}\{\backslash Gamma\{\backslash left(\backslash tfrac\{d\_2\}\{2\}\backslash right)\}\}\; U\; \backslash !\; \backslash left(\backslash frac\{d\_1\}\{2\},1-\backslash frac\{d\_2\}\{2\},-\backslash frac\{d\_2\}\{d\_1\}\; \backslash imath\; s\; \backslash right)$$

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

In instances where the F-distribution is used, for example in the analysis of variance, independence of $U\_1$ and $U\_2$ (defined above) might be demonstrated by applying Cochran's theorem.

Equivalently, since the chi-squared distribution is the sum of squares of independent standard normal random variables, the random variable of the F-distribution may also be written

$$X\; =\; \backslash frac\{s\_1^2\}\{\backslash sigma\_1^2\}\; \backslash div\; \backslash frac\{s\_2^2\}\{\backslash sigma\_2^2\},$$

where $s\_1^2\; =\; \backslash frac\{S\_1^2\}\{d\_1\}$ and $s\_2^2\; =\; \backslash frac\{S\_2^2\}\{d\_2\}$, $S\_1^2$ is the sum of squares of $d\_1$ random variables from normal distribution $N(0,\backslash sigma\_1^2)$ and $S\_2^2$ is the sum of squares of $d\_2$ random variables from normal distribution $N(0,\backslash sigma\_2^2)$.

In a frequentist context, a scaled F-distribution therefore gives the probability $p(s\_1^2/s\_2^2\; \backslash mid\; \backslash sigma\_1^2,\; \backslash sigma\_2^2)$, with the F-distribution itself, without any scaling, applying where $\backslash sigma\_1^2$ is being taken equal to $\backslash sigma\_2^2$. 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 is taken for the prior probabilities of $\backslash sigma\_1^2$ and $\backslash sigma\_2^2$.{{cite book |first=G. E. P. |last=Box |first2=G. C. |last2=Tiao |year=1973 |title=Bayesian Inference in Statistical Analysis |publisher=Addison-Wesley |page=110 |isbn=0-201-00622-7 }} In this context, a scaled F-distribution thus gives the posterior probability $p(\backslash sigma^2\_2\; /\backslash sigma\_1^2\; \backslash mid\; s^2\_1,\; s^2\_2)$, where the observed sums $s^2\_1$ and $s^2\_2$ are now taken as known.

• If $X\; \backslash sim\; \backslash chi^2\_\{d\_1\}$ and $Y\; \backslash sim\; \backslash chi^2\_\{d\_2\}$ (Chi squared distribution) are independent, then $\backslash frac\{X\; /\; d\_1\}\{Y\; /\; d\_2\}\; \backslash sim\; \backslash mathrm\{F\}(d\_1,\; d\_2)$
• If $X\_k\; \backslash sim\; \backslash Gamma(\backslash alpha\_k,\backslash beta\_k)\backslash ,$ (Gamma distribution) are independent, then $\backslash frac\{\backslash alpha\_2\backslash beta\_1\; X\_1\}\{\backslash alpha\_1\backslash beta\_2\; X\_2\}\; \backslash sim\; \backslash mathrm\{F\}(2\backslash alpha\_1,\; 2\backslash alpha\_2)$
• If $X\; \backslash sim\; \backslash operatorname\{Beta\}(d\_1/2,d\_2/2)$ (Beta distribution) then $\backslash frac\{d\_2\; X\}\{d\_1(1-X)\}\; \backslash sim\; \backslash operatorname\{F\}(d\_1,d\_2)$
• Equivalently, if $X\; \backslash sim\; F(d\_1,\; d\_2)$, then $\backslash frac\{d\_1\; X/d\_2\}\{1+d\_1\; X/d\_2\}\; \backslash sim\; \backslash operatorname\{Beta\}(d\_1/2,d\_2/2)$.
• If $X\; \backslash sim\; F(d\_1,\; d\_2)$, then $\backslash frac\{d\_1\}\{d\_2\}X$ has a beta prime distribution: $\backslash frac\{d\_1\}\{d\_2\}X\; \backslash sim\; \backslash operatorname\{\backslash beta^\backslash prime\}\backslash left(\backslash tfrac\{d\_1\}\{2\},\backslash tfrac\{d\_2\}\{2\}\backslash right)$.
• If $X\; \backslash sim\; F(d\_1,\; d\_2)$ then $Y\; =\; \backslash lim\_\{d\_2\; \backslash to\; \backslash infty\}\; d\_1\; X$ has the chi-squared distribution $\backslash chi^2\_\{d\_1\}$
• $F(d\_1,\; d\_2)$ is equivalent to the scaled Hotelling's T-squared distribution $\backslash frac\{d\_2\}\{d\_1(d\_1+d\_2-1)\}\; \backslash operatorname\{T\}^2\; (d\_1,\; d\_1\; +d\_2-1)$.
• If $X\; \backslash sim\; F(d\_1,\; d\_2)$ then $X^\{-1\}\; \backslash sim\; F(d\_2,\; d\_1)$.
• If $X\backslash sim\; t\_\{(n)\}$ — Student's t-distribution — then: $$\backslash begin\{align\}$$

X^{2} &\sim \operatorname{F}(1, n) \\

X^{-2} &\sim \operatorname{F}(n, 1)

\end{align}

• F-distribution is a special case of type 6 Pearson distribution
• If $X$ and $Y$ are independent, with $X,Y\backslash sim$ Laplace(μ, b) then $$\backslash frac$$

  X-\mu

  Y-\mu

  \sim \operatorname{F}(2,2)
• If $X\backslash sim\; F(n,m)$ then $\backslash tfrac\{\backslash log\{X\}\}\{2\}\; \backslash sim\; \backslash operatorname\{FisherZ\}(n,m)$ (Fisher's z-distribution)
• The noncentral F-distribution simplifies to the F-distribution if $\backslash lambda=0$.
• The doubly noncentral F-distribution simplifies to the F-distribution if $\backslash lambda\_1\; =\; \backslash lambda\_2\; =\; 0$
• If $\backslash operatorname\{Q\}\_X(p)$ is the quantile p for $X\backslash sim\; F(d\_1,d\_2)$ and $\backslash operatorname\{Q\}\_Y(1-p)$ is the quantile $1-p$ for $Y\backslash sim\; F(d\_2,d\_1)$, then $$\backslash operatorname\{Q\}\_X(p)=\backslash frac\{1\}\{\backslash operatorname\{Q\}\_Y(1-p)\}.$$
• F-distribution is an instance of ratio distributions
• W-distribution{{Cite journal |last1=Mahmoudi |first1=Amin |last2=Javed |first2=Saad Ahmed |date=October 2022 |title=Probabilistic Approach to Multi-Stage Supplier Evaluation: Confidence Level Measurement in Ordinal Priority Approach |journal=Group Decision and Negotiation |language=en |volume=31 |issue=5 |pages=1051–1096 |doi=10.1007/s10726-022-09790-1 |issn=0926-2644 |pmc=9409630 |pmid=36042813}} is a unique parametrization of F-distribution.

{{Colbegin}}

• Beta prime distribution
• Chi-square distribution
• Chow test
• Gamma distribution
• Hotelling's T-squared distribution
• Wilks' lambda distribution
• Wishart distribution
• Modified half-normal distribution{{cite journal |last1=Sun |first1=Jingchao |last2=Kong |first2=Maiying |last3=Pal |first3=Subhadip |title=The Modified-Half-Normal distribution: Properties and an efficient sampling scheme |journal=Communications in Statistics - Theory and Methods |date=22 June 2021 |volume=52 |issue=5 |pages=1591–1613 |doi=10.1080/03610926.2021.1934700 |s2cid=237919587 |url=https://figshare.com/articles/journal_contribution/The_Modified-Half-Normal_distribution_Properties_and_an_efficient_sampling_scheme/14825266/1/files/28535884.pdf |issn=0361-0926}} with the pdf on $(0,\; \backslash infty)$ is given as $f(x)=\; \backslash frac\{2\backslash beta^\{\backslash frac\{\backslash alpha\}\{2\}\}\; x^\{\backslash alpha-1\}\; \backslash exp(-\backslash beta\; x^2+\; \backslash gamma\; x\; )\}\{\backslash Psi\{\backslash left(\backslash frac\{\backslash alpha\}\{2\},\; \backslash frac\{\; \backslash gamma\}\{\backslash sqrt\{\backslash beta\}\}\backslash right)\}\}$, where $\backslash Psi(\backslash alpha,z)=\{\}\_1\backslash Psi\_1\backslash left(\backslash begin\{matrix\}\backslash left(\backslash alpha,\backslash frac\{1\}\{2\}\backslash right)\backslash \backslash (1,0)\backslash end\{matrix\};z\; \backslash right)$ denotes the Fox–Wright Psi function.

{{Colend}}

{{reflist}}

• [http://www.itl.nist.gov/div898/handbook/eda/section3/eda3673.htm Table of critical values of the F-distribution]
• [https://mathshistory.st-andrews.ac.uk/Miller/mathword/f/ Earliest Uses of Some of the Words of Mathematics: entry on F-distribution contains a brief history]
• [http://www.waterlog.info/f-test.htm Free calculator for F-testing]

{{ProbDistributions|continuous-semi-infinite}}

{{DEFAULTSORT:F-distribution}}

Category:Continuous distributions

Category:Analysis of variance