Applied Statistics and Probability for Engineers

356 CHAPTER 10 STATISTICAL INFERENCE FOR TWO SAMPLES

independent chi-square random variables, each divided by its number of degrees of free-

dom. That is,

(10-25)

where Wand Yare independent chi-square random variables with uand vdegrees of freedom,

respectively. We now formally state the sampling distribution of F.

F

W u

Y v

Let Wand Ybe independent chi-square random variables with uand vdegrees of

freedom, respectively. Then the ratio

(10-26)

has the probability density function

(10-27)

and is said to follow the Fdistribution with udegrees of freedom in the numerator

and vdegrees of freedom in the denominator. It is usually abbreviated as Fu,v.

f 1 x 2 

 a

u v

2

ba

u

vb

u 

2

x^1 u^22 ^1

 a

u

2

b  a

v

2

b c a

u

vb^ x^1 d

1 u v 2

2 ,^0 x

F

W u

Y v

Definition

The mean and variance of the Fdistribution are v(v2) for v 2, and

Tw o Fdistributions are shown in Fig. 10-4. The Frandom variable is nonnegative, and the

distribution is skewed to the right. The Fdistribution looks very similar to the chi-square dis-

tribution; however, the two parameters uand vprovide extra flexibility regarding shape.

The percentage points of the Fdistribution are given in Table V of the Appendix. Let

f ,u,vbe the percentage point of the Fdistribution, with numerator degrees of freedom uand

denominator degrees of freedom vsuch that the probability that the random variable Fex-

ceeds this value is

This is illustrated in Fig. 10-5. For example, if u5 and v10, we find from Table V of the

Appendix that

P 1 F f0.05,5,10 2 P 1 F5,10 3.33 2 0.05

P 1 F f , u, v 2  



f ,u,v

f 1 x 2 dx

^2 

2 v^21 u v 22

u 1 v 2221 v 42

, v

4

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