Page 151 - Fundamentals of Probability and Statistics for Engineers

134 Fundamentals of Probability and Statistics for Engineers

where
g V 1 †ˆ 0; V 1 < 95;

V 1
95
g V 1 †ˆ ; 95 V 1 105;
10
g V 1 †ˆ 1; V 1 > 105:

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

95†
P V 2 ˆ 0†ˆ P V 1 95†ˆ F V 1
Z 95 1
ˆ f V 1 v 1 †dv 1 ˆ ;
90 4
105†
P V 2 ˆ 1†ˆ P V 1 > 105†ˆ 1
F V 1
1
ˆ :
4
For the middle portion, Equation (5.7) leads to

1
‰g v 2 †Š
F V 2 v 2 †ˆ F V 1
10v 2 ‡ 95†; 0 < v 2 < 1:
ˆ F V 1
Now,

v 1
90
v 1 †ˆ ; 90 v 1 110:
F V 1
20
We thus have

1 2v 2 ‡ 1
v 2 †ˆ 10v 2 ‡ 95
90†ˆ ; 0 < v 2 < 1:
F V 2
20 4
(v 2 ), is shown in Figure 5.15, an example of a mixed distribution.
The PDF, F V 2

5.1.2 MOMENTS

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

TLFeBOOK