Mathematical Methods for Physics and Engineering - Matematica.NET

STATISTICSdistribution P (v 2 ,F|H 0 ) is obtained by requiring thatP (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 of F alone, we now integrate P (v 2 ,F|H 0 ) withrespect to v 2 from 0 to ∞, from which we obtainP (F|H 0 )(N1 − 1=N 2 − 1) (N1−1)/2B ( 1F (N1−3)/2 (2 (N 1 − 1), 1 2 (N 2 − 1) )1+ N ) −(N1+N1 − 12−2)/2N 2 − 1 F ,(31.126)where B ( 12 (N 1 − 1), 1 2 (N 2 − 1) ) is the beta function defined in the Appendix.P (F|H 0 ) is called the F-distribution (or occasionally the Fisher distribution) with(N 1 − 1, N 2 − 1) degrees of freedom.◮Evaluate the integral ∫ ∞0 P (v2 ,F|H 0 ) d(v 2 ) to obtain result (31.126).From (31.125), we haveP (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 2d(v 2 ).2σ 2Making the substitution x =[(N 1 − 1)F +(N 2 − 1)]v 2 /(2σ 2 ), we obtain[P (F|H 0 )=A2σ 2(N 1 − 1)F +(N 2 − 1)[2σ 2= A(N 1 − 1)F +(N 2 − 1)] (N1 +N 2 −2)/2F (N 1−3)/2∫ ∞0x (N 1+N 2 −4)/2 e −x dx] (N1 +N 2 −2)/2F (N 1−3)/2 Γ ( 12 (N 1 + N 2 − 2) ) ,where in the last line we have used the definition of the gamma function given in theAppendix. Using the further result (18.165), which expresses the beta function in terms ofthe gamma function, and the expression for A given in (31.124), we see that P (F|H 0 )isindeed given by (31.126). ◭As it does not matter whether the ratio F given in (31.123) is defined as u 2 /v 2or as v 2 /u 2 , it is conventional to put the larger sample variance on the top, sothat F is always greater than or equal to unity. A large value of F indicates thatthe sample variances u 2 and v 2 are very different whereas a value of F close tounity means that they are very similar. Therefore, for a given significance α, itis1294