Mathematical Methods for Physics and Engineering : A Comprehensive Guide

STATISTICS

distributionP(v^2 ,F|H 0 ) is obtained by requiring that

P(v^2 ,F|H 0 )d(v^2 )dF=P(u^2 |H 0 )P(v^2 |H 0 )d(u^2 )d(v^2 ).

(31.125)

In order to find the distribution ofFalone, we now integrateP(v^2 ,F|H 0 ) with

respect tov^2 from 0 to∞, from which we obtain

P(F|H 0 )

=

(

N 1 − 1

N 2 − 1

)(N 1 −1)/ 2

F(N^1 −3)/^2

B

( 1

2 (N^1 −1),

1

2 (N^2 −1)

)

(

1+

N 1 − 1

N 2 − 1

F

)−(N 1 +N 2 −2)/ 2

,

(31.126)

whereB

( 1

2 (N^1 −1),

1

2 (N^2 −1)

)

is the beta function defined in the Appendix.

P(F|H 0 ) is called theF-distribution(or occasionally theFisher distribution) with

(N 1 − 1 ,N 2 −1) degrees of freedom.

Evaluate the integral

∫∞

0 P(v

(^2) ,F|H 0 )d(v (^2) )to obtain result (31.126).
From (31.125), we have

P(F|H 0 )=AF(N^1 −3)/^2

∫∞

0

(v^2 )(N^1 +N^2 −4)/^2 exp

{

−

[(N 1 −1)F+(N 2 −1)]v^2

2 σ^2

}

d(v^2 ).

Making the substitutionx=[(N 1 −1)F+(N 2 −1)]v^2 /(2σ^2 ), we obtain

P(F|H 0 )=A

[

2 σ^2

(N 1 −1)F+(N 2 −1)

](N 1 +N 2 −2)/ 2

F(N^1 −3)/^2

∫∞

0

x(N^1 +N^2 −4)/^2 e−xdx

=A

[

2 σ^2

(N 1 −1)F+(N 2 −1)

](N 1 +N 2 −2)/ 2

F(N^1 −3)/^2 Γ

( 1

2 (N^1 +N^2 −2)

)

,

where in the last line we have used the definition of the gamma function given in the
Appendix. Using the further result (18.165), which expresses the beta function in terms of
the gamma function, and the expression forAgiven in (31.124), we see thatP(F|H 0 )is
indeed given by (31.126).

As it does not matter whether the ratioFgiven in (31.123) is defined asu^2 /v^2

or asv^2 /u^2 , it is conventional to put the larger sample variance on the top, so

thatFis always greater than or equal to unity. A large value ofFindicates that

the sample variancesu^2 andv^2 are very different whereas a value ofFclose to

unity means that they are very similar. Therefore, for a given significanceα,itis