Page 77 - Fundamentals of Probability and Statistics for Engineers

60 Fundamentals of Probability and Statistics for Engineers

Similarly,
1
8
; for 0 y 60;
<
f y†ˆ 60
Y
:
0; elsewhere:
Both random variables are thus uniformly distributed over the interval (0, 60).

Example 3.8. In structural reliability studies, the resistance Y of a structural

element and the force X applied to it are generally regarded as random vari-
ables. The probability of failure, p f , is defined by P(Y X ). Suppose that the
jpdf of X and Y is specified to be
ax‡by†
abe ; for x; y† > 0;
f x; y†ˆ
XY
0; for x; y† 0;
where a and b are known positive constants, we wish to determine p f .
The probability p f is determined from

Z Z
p f ˆ f XY x; y†dxdy;
R
where R is the region satisfying Y X . Since X and Y take only positive values,
the region R is that shown in Figure 3.17. Hence,
1
Z Z 1 b
p f ˆ abe ax‡by† dxdy ˆ :
0 y a ‡ b

y
x = y

R

x

Figure 3.17 Region R in Example 3.8

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