Introduction to Probability and Statistics for Engineers and Scientists

192 Chapter 5: Special Random Variables

following equalities:

α=P

{

χn^2 /n

χm^2 /m

>Fα,n,m

}

=P

{

χm^2 /m

χn^2 /n

<

1

Fα,n,m

}

= 1 −P

{

χm^2 /m

χn^2 /n

≥

1

Fα,n,m

}

or, equivalently,

P

{

χm^2 /m

χn^2 /n

≥

1

Fα,n,m

}

= 1 −α (5.8.3)

But because (χm^2 /m)/(χn^2 /n) has anF-distribution with degrees of freedommandn,it
follows that

1 −α=P

{

χm^2 /m

χn^2 /n

≥F 1 −α,m,n

}

implying, from Equation 5.8.3, that

1

Fα,n,m

=F 1 −α,m,n

Thus, for instance,F.9,5,7=1/F.1,7,5=1/3.37=.2967 where the value ofF.1,7,5was
obtained from Table A4 of the Appendix.
Program 5.8.3 computes the distribution function ofFn,m.

EXAMPLE 5.8f DetermineP{F6,14≤1. 5}.

SOLUTION Run Program 5.8.3 to obtain the solution .7518. ■

*5.9The Logistics Distribution

A random variableXis said to have alogisticsdistribution with parametersμandv>0if
its distribution function is

F(x)=

e(x−μ)/v

1 +e(x−μ)/v

, −∞<x<∞

* Optional section.