Page 209 - Fundamentals of Probability and Statistics for Engineers

192 Fundamentals of Probability and Statistics for Engineers

f (x) F (x)
X
X

1
1
b – a

x x
a b a b
(a) (b)

Figure 7.1 (a) The probability density function, f (x), and (b) the probability

X
distribution function, F X (x), of X
which is graphically presented in Figure 7.1(b).
The mean, m X , and variance, 2 , of X are easily found to be
X
b b
Z 1 Z a ‡ b
m X ˆ xf x†dx ˆ xdx ˆ ;
X
a b a a 2
7:3†
b
1 Z a ‡ b 2 b a† 2
2
ˆ x dx ˆ :
X
b a a 2 12
The uniform distribution is one of the simplest distributions and is com-
monly used in situations where there is no reason to give unequal likelihoods to
possible ranges assumed by the random variable over a given interval. For
example, the arrival time of a flight might be considered uniformly distributed
over a certain time interval, or the distribution of the distance from the location
of live loads on a bridge to an end support might be adequately represented by
a uniform distribution over the bridge span. Let us also comment that one often
assigns a uniform distribution to a specific random variable simply because of
a lack of information, beyond knowing the range of values it spans.
Example 7.1. Problem: owing to unpredictable traffic situations, the time
required by a certain student to travel from her home to her morning class
is uniformly distributed between 22 and 30 minutes. If she leaves home at pre-
cisely 7.35 a.m., what is the probability that she will not be late for class, which
begins promptly at 8:00 a.m.?
Answer: let X be the class arrival time of the student in minutes after 8:00 a.m.
It then has a uniform distribution given by
1
8
; for 3 x 5;
<
f x†ˆ 8
X
0; elsewhere.
:

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