Page 261 - Fundamentals of Probability and Statistics for Engineers

244 Fundamentals of Probability and Statistics for Engineers

s

1 2 .... n

s
Figure 7.17 Structure under load s, for Problem 7.32

7.33 What is the probability sought in Problem 7.32 if the load is also a random variable
S with pdf f (s)?
S
7.34 Let n ˆ 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 ˆ 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-
0 5, 1, 2, and 3; and w ˆ
ing combinations of parameter values: k ˆ : 1 and 2.
7.36 The ranges of n independent test flights of a supersonic aircraft are assumed to be
identically distributed with PDF F X (x) and pdf f (x). If range span is defined as the
X
distance between the maximum and minimum ranges of these n values, determine
the pdf of the range span in terms of F X (x) or f (x). Expressing it mathematically,
X
the pdf of interest is that of S, where
S ˆ Y Z;

with
Y ˆ max X 1 ; X 2 ; .. . ; X n †;

and
Z ˆ min X 1 ; X 2 ; .. . ; X n †:
Note that random variables Y and Z are not independent.

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