F-distribution

Properties

The expectation, variance, and other details about the F-distribution ⁠${\displaystyle F(d_1,d_2)}$⁠ are given in the sidebox; for ⁠${\displaystyle d_2>8}$⁠, the excess kurtosis is $${\displaystyle {\gamma }_2=12\frac{d_1(5d_2-22)(d_1+d_2-2)+(d_2-4)(d_2-2{)}^2}{d_1(d_2-6)(d_2-8)(d_1+d_2-2)}.}$$

The k-th moment of an ⁠${\displaystyle F(d_1,d_2)}$⁠ distribution exists and is finite only when ⁠${\displaystyle 2k<d_2}$⁠ and it is equal to

$${\displaystyle {\mu }_X(k)={\left(\frac{d_2}{d_1}\right)}^k\frac{\mathrm{\Gamma }\left(\frac{d_1}{2}+k\right)}{\mathrm{\Gamma }\left(\frac{d_1}{2}\right)}\frac{\mathrm{\Gamma }\left(\frac{d_2}{2}-k\right)}{\mathrm{\Gamma }\left(\frac{d_2}{2}\right)}.}$$

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 is

$${\displaystyle {\phi }_{d_1,d_2}^F(s)=\frac{\mathrm{\Gamma }\left(\frac{d_1+d_2}{2}\right)}{\mathrm{\Gamma }\left(\frac{d_2}{2}\right)}U\left(\frac{d_1}{2},1-\frac{d_2}{2},-\frac{d_2}{d_1}ıs\right)}$$

where ⁠${\displaystyle U(a,b,z)}$⁠ is the confluent hypergeometric function of the second kind.

Related distributions

Relation to the chi-squared distribution

In instances where the F-distribution is used, for example in the analysis of variance, independence of ⁠${\displaystyle U_1}$⁠ and ⁠${\displaystyle U_2}$⁠ (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 $${\displaystyle X=\frac{s_1^2}{{\sigma }_1^2}÷\frac{s_2^2}{{\sigma }_2^2},}$$

where $s_1^2=\frac{S_1^2}{d_1}$ and ⁠${\displaystyle s_2^2=\frac{S_2^2}{d_2}}$⁠, ${\displaystyle S_1^2}$ is the sum of squares of ${\displaystyle d_1}$ random variables from normal distribution ${\displaystyle N(0,{\sigma }_1^2)}$ and ${\displaystyle S_2^2}$ is the sum of squares of ${\displaystyle d_2}$ random variables from normal distribution ⁠${\displaystyle N(0,{\sigma }_2^2)}$⁠.

In a frequentist context, a scaled F-distribution therefore gives the probability ⁠${\displaystyle p(s_1^2/s_2^2\mid {\sigma }_1^2,{\sigma }_2^2)}$⁠, with the F-distribution itself, without any scaling, applying where ${\displaystyle {\sigma }_1^2}$ is being taken equal to ⁠${\displaystyle {\sigma }_2^2}$⁠. 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 ${\displaystyle X}$ has the same distribution in Bayesian statistics, if an uninformative rescaling-invariant Jeffreys prior is taken for the prior probabilities of ${\displaystyle {\sigma }_1^2}$ and ⁠${\displaystyle {\sigma }_2^2}$⁠. In this context, a scaled F-distribution thus gives the posterior probability ⁠${\displaystyle p({\sigma }_2^2/{\sigma }_1^2\mid s_1^2,s_2^2)}$⁠, where the observed sums ${\displaystyle s_1^2}$ and ${\displaystyle s_2^2}$ are now taken as known.

In general

• If ${\displaystyle X\sim {\chi }_{d_1}^2}$ and ${\displaystyle Y\sim {\chi }_{d_2}^2}$ (Chi squared distribution) are independent, then ⁠${\displaystyle \frac{X/d_1}{Y/d_2}\sim \mathrm{F}(d_1,d_2)}$⁠.
• If ${\displaystyle X_k\sim \mathrm{\Gamma }({\alpha }_k,{\beta }_k)}$ (Gamma distribution) are independent, then ⁠${\displaystyle \frac{{\alpha }_2{\beta }_1{X}_1}{{\alpha }_1{\beta }_2{X}_2}\sim \mathrm{F}(2{\alpha }_1,2{\alpha }_2)}$⁠.
• If $X\sim \mathrm{Beta}\left(\frac{d_1}{2},\frac{d_2}{2}\right)$ (Beta distribution) then ⁠${\displaystyle \frac{d_{2}X}{d_1(1-X)}\sim \mathrm{F}(d_1,d_2)}$⁠.
  • Equivalently, if ⁠${\displaystyle X\sim F(d_1,d_2)}$⁠, then ⁠${\displaystyle \frac{d_{1}X/d_2}{1+d_{1}X/d_2}\sim \mathrm{Beta}\left(\frac{d_1}{2},\frac{d_2}{2}\right)}$⁠.
• If ⁠${\displaystyle X\sim F(d_1,d_2)}$⁠, then $\frac{d_1}{d_2}X$ has a beta prime distribution: ⁠${\displaystyle \frac{d_1}{d_2}X\sim {\beta }^{\mathrm{\prime }}\left(\frac{d_1}{2},\frac{d_2}{2}\right)}$⁠.
• If ${\displaystyle X\sim F(d_1,d_2)}$ then $Y=\underset{d_2\to \mathrm{\infty }}{lim}{d}_{1}X$ has the chi-squared distribution ⁠${\displaystyle {\chi }_{d_1}^2}$⁠.
• ${\displaystyle F(d_1,d_2)}$ is equivalent to the scaled Hotelling's T-squared distribution ⁠${\displaystyle \frac{d_2}{d_1(d_1+d_2-1)}{\mathrm{T}}^2(d_1,d_1+d_2-1)}$⁠.
• If ${\displaystyle X\sim F(d_1,d_2)}$ then ⁠${\displaystyle X^{-1}\sim F(d_2,d_1)}$⁠.
• If ⁠${\displaystyle X\sim t_{(n)}}$⁠ – Student's t-distribution – then: $${\displaystyle \begin{array}{cc}{X}^2& \sim \mathrm{F}(1,n),\\ X^{-2}& \sim \mathrm{F}(n,1).\end{array}}$$
• F-distribution is a special case of type 6 Pearson distribution.
• If ${\displaystyle X}$ and ${\displaystyle Y}$ are independent, with ⁠${\displaystyle X,Y\sim \mathrm{Laplace}(\mu ,b)}$⁠ (Laplace distribution), then $${\displaystyle \frac{|X-\mu |}{|Y-\mu |}\sim \mathrm{F}(2,2).}$$
• If ${\displaystyle X\sim F(n,m)}$ then ${\displaystyle \frac{\log X}{2}\sim \mathrm{FisherZ}(n,m)}$ (Fisher's z-distribution).
• The noncentral F-distribution simplifies to the F-distribution if ⁠${\displaystyle \lambda =0}$⁠.
• The doubly noncentral F-distribution simplifies to the F-distribution if ⁠${\displaystyle {\lambda }_1={\lambda }_2=0}$⁠
• If ${\displaystyle {\mathrm{Q}}_X(p)}$ is the quantile ⁠${\displaystyle p}$⁠ for ${\displaystyle X\sim F(d_1,d_2)}$ and ${\displaystyle {\mathrm{Q}}_Y(1-p)}$ is the quantile ${\displaystyle 1-p}$ for ⁠${\displaystyle Y\sim F(d_2,d_1)}$⁠, then $${\displaystyle {\mathrm{Q}}_X(p)=\frac{1}{{\mathrm{Q}}_Y(1-p)}.}$$
• F-distribution is an instance of ratio distributions.
• W-distribution is a unique parametrization of F-distribution.