Fundamentals of Probability and Statistics for Engineers

which is graphically presented in Figure 7.1(b).
The mean, mX, and variance, , of X are easily found to be

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 theclass arrivaltimeof the student in minutes after 8:00 a.m.
It then has a uniform distribution given by

fX(x)

a

(a) (b)

1

b x x

b – a

FX(x)

a

1

b

Figure 7. 1 (a) The probability density function, fX (x), and (b) the probability

distribution function, FX (x), of X

192 Fundamentals of Probability and Statistics for Engineers

^2 X

mXˆ

Zb

a

xfX…x†dxˆ

1

ba

Zb

a

xdxˆ

a‡b

2

;

^2 Xˆ

1

ba

Zb

a

x

a‡b

2

 2

dxˆ

…ba†^2

12

:

… 7 : 3 †

fX…x†ˆ

1

8

; for 3 x 5 ;

0 ; elsewhere.

8

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