Introduction to Probability and Statistics for Engineers and Scientists

5.8Distributions Arising from the Normal 191

Area = a

−ta, n = t 1 −a, n 0 ta, n

Area = a

FIGURE 5.16 t 1 −α,n=−tα,n.

EXAMPLE 5.8e Find(a)P{T 12 ≤1.4}and(b)t.025,9.

SOLUTION Run Programs 5.8.2a and 5.8.2b to obtain the results.

(a).9066 (b)2.2625 ■

5.8.3 TheF-Distribution..............................................

Ifχn^2 andχm^2 areindependentchi-squarerandomvariableswithnandmdegreesoffreedom,
respectively, then the random variableFn,mdefined by

Fn,m=

χn^2 /n

χm^2 /m

is said to have anF-distribution with n and m degrees of freedom.
For anyα∈(0, 1), letFα,n,mbe such that

P{Fn,m>Fα,n,m}=α

This is illustrated in Figure 5.17.
The quantitiesFα,n,mare tabulated in Table A4 of the Appendix for different values
ofn,m, andα≤^12 .IfFα,n,mis desired whenα>^12 , it can be obtained by using the

0

Area = a

Fa, n, m

FIGURE 5.17 Density function of Fn,m.