F distribution

Let X and Y be random variablesMathworldPlanetmath such that

  1. 1.

    X and Y are independentPlanetmathPlanetmath

  2. 2.

    Xχ2(m), the chi-squared distribution (http://planetmath.org/ChiSquaredRandomVariable) with m degrees of freedom

  3. 3.

    Yχ2(n), the chi-squared distribution with n degrees of freedom

Define a new random variable Z by


Then the distributionDlmfPlanetmathPlanetmath of Z is called the central F distribution, or simply the F distribution with m and n degrees of freedom, denoted by ZF(m,n).

By transformation of the random variables X and Y, one can show that the probability density function of the F distribution of Z has the form:


for x>0, where B(α,β) is the beta functionDlmfDlmfMathworldPlanetmath. fZ(x)=0 for x0.

For a fixed m, say 10, below are some graphs for the probability density functions of the F distribution with (m,n) degrees of freedom.

The next set of graphs shows the density functions with (m,n) degrees of freedom when n is fixed. In this example, n=10.

If Xχ2(m,λ), the non-central chi-square distribution with m degrees of freedom and non-centrality parameter λ, with Y and Z defined as above, then the distribution of Z is called the non-central F distribution with m and n degrees of freedom and non-centrality parameter λ.


  • the “F” in the F distribution is given in honor of statistician R. A. Fisher.

  • If XF(m,n), then 1/XF(n,m).

  • If Xt(n), the t distribution with n degrees of freedom, then X2F(1,n).

  • If XF(m,n), then

    E[X]=nn-2 if n>2,


    Var[X]=2n2(m+n-2)m(n-2)2(n-4) if n>4.
  • Suppose X1,,Xm are random samples from a normal distributionMathworldPlanetmath with mean μ1 and varianceMathworldPlanetmath σ12. Furthermore, suppose Y1,,Yn are random samples from another normal distribution with mean μ2 and variance σ22. Then the statisticMathworldMathworldPlanetmath defined by


    where σ1^2 and σ1^2 are sample variances of the Xis and the Yjs, respectively, has an F distribution with m and n degrees of freedom. V can be used to test whether σ12=σ22. V is an example of an F test.

Title F distribution
Canonical name FDistribution
Date of creation 2013-03-22 14:26:56
Last modified on 2013-03-22 14:26:56
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 15
Author CWoo (3771)
Entry type Definition
Classification msc 62A01
Synonym Fisher F distribution
Synonym F-distribution
Synonym central F-distribution
Synonym central F distribution
Defines non-central F distribution