Page 160 - Fundamentals of Probability and Statistics for Engineers

Functions of Random Variables 143

Table 5.1 Joint probability mass
function, p (x 1 , x 2 ), in Example 5.13
X 1 X 2
x 2 x 1
1 2 3

1 0.04 0.06 0.12
2 0.5 0.24 0.04

Example 5.14. Problem: in structural reliability studies, the probability of
failure q is defined by
q ˆ P R S†;

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

R
q ˆ P 1 ˆ P Y 1†ˆ F Y 1†:
S
Identifying R and S with X 1 and X 2 , respectively, in Example 5.12, it follows
from Equation (5.49) that
Z 1
q ˆ F Y 1†ˆ F R s†f s†ds:
S
0

Example 5.15. Problem: determine F Y (y) in terms of f (x 1 , x 2 ) when

X 1 X 2
Y ˆ min (X 1 , X 2 ).
Answer: now,
ZZ
F Y y†ˆ f x 1 ; x 2 †dx 1 dx 2 ;
X 1 X 2
R : min x 1 ;x 2 † y†
2
2
where region R is shown in Figure 5.19. Thus
y 1 1 y
Z Z Z Z
F Y y†ˆ f x 1 ; x 2 †dx 1 dx 2 ‡ f x 1 ; x 2 †dx 1 dx 2
X 1 X 2 X 1 X 2

1
1 y
1
Z y Z 1 Z 1 Z y
ˆ f x 1 ; x 2 †dx 1 dx 2 ‡ f x 1 ; x 2 †dx 1 dx 2
X 1 X 2 X 1 X 2

1
1
1
1
Z y Z y

f x 1 ; x 2 †dx 1 dx 2
X 1 X 2

1
1
y; y†;
ˆ F X 2 y†‡ F X 1 y†
F X 1 X 2

TLFeBOOK