Fundamentals of Probability and Statistics for Engineers

where

The theorems stated in this section do not apply in this case to the portions
v 1 <95Vandv 1 > 105V because in finite and noncountable number of roots
for v 1 exist in these regions. However, we deduce immediately from Figure 5.14
that

For the middle portion, Equation (5.7) leads to

Now,

We thus have

ThePDF, FV 2 (v 2 ), is shown in Figure 5.15, an example of a mixed distribution.

5.1.2 M oments

Having developed methods of determining the probability distribution of
Y g(X), it is a straightforward matter to calculate all the desired moments

134 Fundamentals of Probability and Statistics for Engineers

g…V 1 †ˆ 0 ; V 1 < 95 ;

g…V 1 †ˆ

V 1

95

10

; 95 V 1  105 ;

g…V 1 †ˆ 1 ; V 1 > 105 :

P…V 2 ˆ 0 †ˆP…V 1  95 †ˆFV 1 … 95 †

ˆ

Z 95

90

fV 1 …v 1 †dv 1 ˆ

1

4

;

P…V 2 ˆ 1 †ˆP…V 1 > 105 †ˆ 1

FV 1 … 105 †

ˆ

1

4

:

FV 2 …v 2 †ˆFV 1 ‰g^1 …v 2 †Š

ˆFV 1 … 10 v 2 ‡ 95 †; 0 <v 2 < 1 :

FV 1 …v 1 †ˆ

v 1

90

20

; 90 v 1  110 :

FV 2 …v 2 †ˆ

1

20

… 10 v 2 ‡ 95

90 †ˆ

2 v 2 ‡ 1

4

; 0 <v 2 < 1 :

ˆ