then examined to see whether their ratio is significantly incompatible with this null hypothesis. The quantity has the same distribution in Bayesian statistics, if an uninformative rescaling-invariant Jeffreys prior is taken for the prior probabilities of and . In this context, a scaled F-distribution thus gives the posterior probability , where the observed sums and are now taken as known. Properties and related distributions If and (Chi squared distribution) are independent, then If (Gamma distribution) are independent, then If (Beta distribution) then Equivalently, if , then . If , then has a beta prime distribution: . If then has the chi-squared distribution is equivalent to the scaled Hotelling's T-squared distribution . If then . If — Student's t-distribution — then: F-distribution is a special case of type 6 Pearson distribution If and are independent, with Laplace(μ, b) then If then (Fisher's z-distribution) The noncentral F-distribution simplifies to the F-distribution if . The doubly noncentral F-distribution simplifies to the F-distribution if If is the quantile p for and is the quantile for , then F-distribution is an instance of ratio distributions See also Beta prime distribution Chi-square distribution Chow test Gamma distribution Hotelling's T-squared distribution Wilks' lambda distribution Wishart distribution Modified half-normal distribution with the pdf on is given as , where denotes the Fox-Wright Psi function. References External links Table of critical values of the F-distribution Earliest Uses of Some of the Words of Mathematics: entry on F-distribution contains a brief history Free calculator for F-testing Continuous distributions Analysis of variance