In probability theory and statistics, the F-distribution or F-ratio, also known as Snedecor's F distribution or the Fisher–Snedecor distribution (after Ronald Fisher and George W. Snedecor) is a continuous probability distribution that arises frequently as the null distribution of a test statistic, most notably in the analysis of variance (ANOVA) and other F-tests. Definition The F-distribution with d1 and d2 degrees of freedom is the distribution of where and are independent random variables with chi-square distributions with respective degrees of freedom and . It can be shown to follow that the probability density function (pdf) for X is given by for real x > 0. Here 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 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 The k-th moment of an F(d1, d2) distribution exists and is finite only when 2k < d2 and it is equal to 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 where U(a, b, z) is the confluent hypergeometric function of the second kind. Characterization A random variate of the F-distribution with parameters and arises as the ratio of two appropriately scaled chi-squared variates: where and have chi-squared distributions with and degrees of freedom respectively, and and are independent. In instances where the F-distribution is used, for example in the analysis of variance, independence of and might be demonstrated by applying Cochran's theorem. Equivalently, the random variable of the F-distribution may also be written where and , is the sum of squares of random variables from normal distribution and is the sum of squares of random variables from normal distribution . In a frequentist context, a scaled F-distribution therefore gives the probability , with the F-distribution itself, without any scaling, applying where is being taken equal to . 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