FRatioDistribution
FRatioDistribution[n,m]
represents an Fratio distribution with n numerator and m denominator degrees of freedom.
Details
 FRatioDistribution is also known as the Fisher–Snedecor distribution.
 The probability density for value in an Fratio distribution is proportional to for , and is zero for . »
 For integers n and m, the Fratio distribution gives the distribution of the ratio of variances for samples from normal distributions.
 FRatioDistribution allows n and m to be any positive real numbers.
 FRatioDistribution allows n and m to be dimensionless quantities. »
 FRatioDistribution can be used with such functions as Mean, CDF, and RandomVariate. »
Background & Context
 FRatioDistribution[n,m] represents a continuous statistical distribution over the interval defined as the distribution of the ratio X=Y_{1}/Y_{2}, where Y_{1}ChiSquareDistribution[n] and Y_{2}ChiSquareDistribution[m] are independent variates with n and m degrees of freedom, respectively. Depending on the values of n and m, the probability density function (PDF) may be either unimodal or monotonically decreasing with a potential singularity nearing the lower endpoint of its domain. In addition, the tails of the PDF are "fat" in the sense that the PDF decreases algebraically rather than exponentially for large values of . (This behavior can be made quantitatively precise by analyzing the SurvivalFunction of the distribution.) The Fratio distribution is sometimes called the Fisher–Snedecor distribution, (central) Fdistribution, or Snedecor's Fdistribution.
 The Fratio distribution was first formalized in the mid1930s by American mathematician G. W. Snedecor as a tool to improve the analysis of variance as introduced by English statistician R. A. Fisher in the late 1910s. The Fratio distribution is a staple in modern statistics, where it forms the basis for the socalled Ftest. The Ftest is a test whose null hypothesis involves an Fratio distributed test statistic. It is used to test a number of common hypotheses, examples of which include determining if a proposed regression model fits a certain dataset and if the means of normally distributed populations with the same standard deviation are equal. Because of the generality of such tests, the Fratio distribution is used in many fields, including economics, psychology, pharmacy, engineering, medicine, and manufacturing.
 RandomVariate can be used to give one or more machine or arbitraryprecision (the latter via the WorkingPrecision option) pseudorandom variates from an Fratio distribution. Distributed[x,FRatioDistribution[n,m]], written more concisely as xFRatioDistribution[n,m], can be used to assert that a random variable x is distributed according to an Fratio distribution. Such an assertion can then be used in functions such as Probability, NProbability, Expectation, and NExpectation.
 The probability density and cumulative distribution functions may be given using PDF[FRatioDistribution[n,m],x] and CDF[FRatioDistribution[n,m],x]. The mean, median, variance, raw moments, and central moments may be computed using Mean, Median, Variance, Moment, and CentralMoment, respectively.
 DistributionFitTest can be used to test if a given dataset is consistent with an Fratio distribution, EstimatedDistribution to estimate an Fratio parametric distribution from given data, and FindDistributionParameters to fit data to an Fratio distribution. ProbabilityPlot can be used to generate a plot of the CDF of given data against the CDF of a symbolic Fratio distribution and QuantilePlot to generate a plot of the quantiles of given data against the quantiles of a symbolic Fratio distribution.
 TransformedDistribution can be used to represent a transformed Fratio distribution, CensoredDistribution to represent the distribution of values censored between upper and lower values, and TruncatedDistribution to represent the distribution of values truncated between upper and lower values. CopulaDistribution can be used to build higherdimensional distributions that contain an Fratio distribution, and ProductDistribution can be used to compute a joint distribution with independent component distributions involving Fratio distributions.
 The Fratio distribution is related to a number of other distributions. In particular, as previously noted, FRatioDistribution is defined in terms of the ChiSquareDistribution (and hence related to ChiDistribution). Moreover, the PDF of FRatioDistribution[1,m] tends to that of ChiSquareDistribution[1] as m tends to Infinity. FRatioDistribution can be obtained from other distributions both by simple transformations, e.g. as the square of StudentTDistribution, as well as from more complicated transformations of distributions such as LaplaceDistribution and FisherZDistribution. FRatioDistribution is a special case of NoncentralFRatioDistribution, in the sense that the PDF of FRatioDistribution[n,m] is precisely that of both NoncentralFRatioDistribution[n,m,0] and NoncentralFRatioDistribution[n,m,0,0] and is also related to BetaDistribution, PearsonDistribution, HotellingTSquareDistribution, and BinomialDistribution.
Examples
open all close allBasic Examples (4)
Scope (8)
Applications (1)
Properties & Relations (13)
Possible Issues (2)
Neat Examples (1)
Introduced in 2007
Updated in 2016
(6.0)

(10.4)