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F distribution∗
CWoo†
2013-03-21 17:35:07
Let X and Y be random variables such that
1. X and Y are independent
2. X ∼ χ2 (m), the chi-squared distribution with m degrees of freedom
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/2 nn/2
x(m/2)−1
·
,
m n
B( 2 , 2 ) (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.
∗ hFDistributioni created: h2013-03-21i by: hCWooi version: h35964i Privacy setting:
h1i hDefinitioni h62A01i
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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
• the “F” in the F distribution is given in honor of statistician R. A. Fisher.
• If X ∼ F(m, n), then 1/X ∼ F(n, m).
• If X ∼ t(n), the t distribution with n degrees of freedom, then X 2 ∼
F(1, n).
• If X ∼ F(m, n), then
E[X] =
and
Var[X] =
n
if n > 2,
n−2
2n2 (m + n − 2)
if n > 4.
m(n − 2)2 (n − 4)
• Suppose X1 , . . . , Xm are random samples from a normal distribution with
mean µ1 and variance σ12 . Furthermore, suppose Y1 , . . . , Yn are random
samples from another normal distribution with mean µ2 and variance σ22 .
Then the statistic defined by
V =
σˆ1 2
,
σˆ2 2
where σˆ1 2 and σˆ1 2 are sample variances of the Xi0 s and the Yj0 s, 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.
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