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
qP
RS;
qP
R
S
1
P Y 1 FY 1 :
qFY
1
Z 1
0
FR
sfS
sds:
FY
y
ZZ
R^2 :min
x 1 ;x 2 y
fX 1 X 2
x 1 ;x 2 dx 1 dx 2 ;
FY
y
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
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
FX 2
yFX 1
y
FX 1 X 2
y;y;
Y min(X,X