F distribution
Let X and Y be random variables such that
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1.
X and Y are independent
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2.
X∼χ2(m), the chi-squared distribution (http://planetmath.org/ChiSquaredRandomVariable) with m degrees of freedom
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3.
Y∼χ2(n), the chi-squared distribution with n degrees of freedom
Define a new random variable Z by
Z=(X/m)(Y/n). |
Then the distribution of Z is called the central F distribution, or simply the F distribution with m and n degrees of freedom, denoted by Z∼F(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:
fZ(x)=mm/2nn/2B(m2,n2)⋅x(m/2)-1(mx+n)(m+n)/2, |
for x>0, where B(α,β) is the beta function. fZ(x)=0 for x≤0.
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 X∼χ2(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.
Remarks
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•
the “F” in the F distribution is given in honor of statistician R. A. Fisher.
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•
If X∼F(m,n), then 1/X∼F(n,m).
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•
If X∼t(n), the t distribution with n degrees of freedom, then X2∼F(1,n).
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•
If X∼F(m,n), then
E[X]=nn-2 if n>2, and
Var[X]=2n2(m+n-2)m(n-2)2(n-4) if n>4. -
•
Suppose X1,…,Xm are random samples from a normal distribution
with mean μ1 and variance
σ21. Furthermore, suppose Y1,…,Yn are random samples from another normal distribution with mean μ2 and variance σ22. Then the statistic
defined by
V=^σ12^σ22, where ^σ12 and ^σ12 are sample variances of the X′is and the Y′js, respectively, has an F distribution with m and n degrees of freedom. V can be used to test whether σ21=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 |