Fundamentals of Probability and Statistics for Engineers

Example 5.14.Problem: in structural reliability studies, the probability of
failure q is defined by

where R and S represent, respectively, structural resistance and applied force.
Let R and S be in dependent random variables taking only positive values.
Determine q in terms of the probability distributions associated with R and S.
Answer: let Y R/S. Probability q can be expressed by

Identifying R and S with X 1 and X 2 , respectively, in Example 5.12, it follows
from Equation (5.49) that

Ex ample 5. 15. Problem: determine FY (y) in terms of f (^) X 1 X 2 (x 1 ,x 2 )when
1 2 ).
Answer: now,
where region R^2 is shown in Figure 5.19. Thus
Table 5.1 Joint probability mass
function, pX 1 X 2 (x 1 ,x 2 ), in Example 5.13
x 2 x 1
123
1 0.04 0.06 0.12
2 0.5 0.24 0.04
Functions of Random Variables 143
qˆP…RS†;

ˆ

qˆP

R

S

 1



ˆP…Y 1 †ˆFY… 1 †:

qˆFY… 1 †ˆ

Z 1

0

FR…s†fS…s†ds:

FY…y†ˆ

ZZ

…R^2 :min…x 1 ;x 2 †y†

fX 1 X 2 …x 1 ;x 2 †dx 1 dx 2 ;

FY…y†ˆ

Zy

1

Z 1

1

fX 1 X 2 …x 1 ;x 2 †dx 1 dx 2 ‡

Z 1

y

Zy

1

fX 1 X 2 …x 1 ;x 2 †dx 1 dx 2

ˆ

Zy

1

Z 1

1

fX 1 X 2 …x 1 ;x 2 †dx 1 dx 2 ‡

Z 1

1

Zy

1

fX 1 X 2 …x 1 ;x 2 †dx 1 dx 2

Zy

1

Zy

1

fX 1 X 2 …x 1 ;x 2 †dx 1 dx 2

ˆFX 2 …y†‡FX 1 …y†

FX 1 X 2 …y;y†;

Y ˆmin(X,X