Fundamentals of Probability and Statistics for Engineers

Similarly,

Both random variables are thus uniformly distributed over the interval (0, 60).

Ex ample 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, pf , is defined by P(Y X). Suppose that the
jp df of X and Y is specified to be

where a and b are known positive constants, we wish to determine pf.
The probability pf is determined from

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,

R

x=y

x

y

Figure 3. 17 Region R in Example 3.8

60 Fundamentals of Probability and Statistics for Engineers

fY…y†ˆ

1

60

; for 0y 60 ;

0 ; elsewhere:

8

<

:



fXY…x;y†ˆ

abe…ax‡by†; for…x;y†> 0 ;

0 ; for…x;y† 0 ;



pfˆ

Z

R

Z

fXY…x;y†dxdy;



pfˆ

Z 1

0

Z 1

y

abe…ax‡by†dxdyˆ

b

a‡b

: