Begin2.DVI

The F-Distribution

The F-distribution has the probability density function

f(x) = fn,m (x) =











Γ

(

m+n

2

)

Γ

(m

2

)

Γ

(n

2

)nn/^2 mm/^2

xn/^2 −^1

(m+nx )(m+n)/^2

, for x > 0

0 , for x < 0

(11 .78)

which is sometimes given in the form

f(x) = fn,m (x) =











Γ

(

m+n

2

)

Γ

(m

2

)

Γ

(n

2

)(n/m )n/^2

xn/^2 −^1

(1 + n

m

x)(m+n)/^2

, for x > 0

0 , for x < 0

(11 .79)

where Γ( ) denotes the gamma function. The F-distribution has the parameters

m= 1, 2 , 3 ,... and n= 1, 2 , 3 ,.. ..

If X 1 and X 2 are independent random variables associated with a chi-square

distribution having respectively the degrees of freedom nand m, then the quantity

Y =

X 1 /n

X 2 /m will have a F-distribution with nand mdegrees of freedom.

The tables 11.6 (a)(b)(c)(d)(e) contain values of Fα,n,m such that

∫∞

F(α,n,m )

fn,m (x)dx =α

for αhaving the values 0. 1 , 0. 05 ,. 025 ,. 01 ,and. 005. Observe the symmetry of the F-

distribution and note that in the use of the upper tail values from the tables it is

customary to employ the relation

F(dfm, dfn, 1 −α/2) =

1

F(dfn, dfm, α/ 2) (11 .80)

where dfmand dfn denote the degrees of freedom for mand n.

The chi-square, student t and F distributions are used in testing of hypothesis,

confidence intervals and testing differences or ratios of various statistics associated

with independent samples. The degrees of freedom associated with these distribu-

tions can be thought of as a parameter representing an increase in reliability of the

calculated statistic. That is, a statistic associated with one degree of freedom is less

reliable than the same statistic calculated using a higher degree of freedom. In some

cases the degrees of freedom are related to the number of data points used to calcu-

late the statistic. In some cases the degrees of freedom is obtained by subtracting 1

from the sample size n.