Page 86 - Fundamentals of Probability and Statistics for Engineers

Random Variables and Probability Distributions 69

f (x)
X

a

x
–3 3

Figure 3.20 The probability density function, f (x), for Problem 3.5
X
3.7 Let T denote the life (in months) of a light bulb and let
1 t
8
< ; for 0 t 30;
f t†ˆ 15 450
T
0; elsewhere:
:
(a) Plot f (t) against t.
T
(b) Derive F T (t) and plot F T (t) against t.
(c) Determine using f (t), the probability that the light bulb will last at least 15
T
months.
(d) Determine, using F T (t), the probability that the light bulb will last at least 15
months.
(e) A light bulb has already lasted 15 months. What is the probability that it will
survive another month?
3.8 The time, in minutes, required for a student to travel from home to a morning
class is uniformly distributed between 20 and 25. If the student leaves home
promptly at 7:38 a.m., what is the probability that the student will not be late for
class at 8:00 a.m.?
3.9 In constructing the bridge shown in Figure 3.21, an engineer is concerned with
forces acting on the end supports caused by a randomly applied concentrated load
P, the term ‘randomly applied’ meaning that the probability of the load lying in any
region is proportional only to the length of that region. Suppose that the bridge has
a span 2b. Determine the PDF and pdf of random variable X, which is the distance
from the load to the nearest edge support. Sketch these functions.

P

2b
Figure 3.21 Diagram of the bridge, for Problem 3.9

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