Fundamentals of Probability and Statistics for Engineers

7.33 What is the probability sought in Problem 7.32 if the load is also a random variable
S with pdf fS (s)?

7.34 Let n 3 in Problem 7.32. Determine the probabilities of failure in zero, one, two,
and three members in Problem 7.32 if R follows a uniform distribution over
interval (80, 100), and s 270. Is partial failure (one-member or two-member
failure) possible in this case?

7.35 To show that, as a time-to-failure model, the Weibull distribution corresponds to
a wide variety of shapes of the hazard function, graph the hazard function in Equation
(7.123) and the corresponding Weibull distribution in Equation (7.124) for the follow-
ing combinations of parameter values: k 0 5, 1, 2, and 3; and w 1 and 2.

7.36 The ranges of n independent test flights of a supersonic aircraft are assumed to be
identically distributed with PDF FX (x) and pdf fX (x). If range span is defined as the
distance between the maximum and minimum ranges of these n values, determine

the pdf of the range span in terms of FX (x) or f (^) X (x). Expressing it mathematically,
the pdf of interest is that of S, where
with
and
Note that random variables Y and Z are not independent.
s
s
1 2 ....n
Figure 7.17 Structure under load s, for Problem 7.32
244 Fundamentals of Probability and Statistics for Engineers
ˆ
ˆ
ˆ : ˆ
SˆYZ;
Yˆmax…X 1 ;X 2 ;...;Xn†;
Zˆmin…X 1 ;X 2 ;...;Xn†: