Page 208 - Fundamentals of Probability and Statistics for Engineers

7

Some Important Continuous

Distributions

Let us turn our attention to some important continuous probability distribu-
tions. Physical quantities such as time, length, area, temperature, pressure, load,
intensity, etc., when they need to be described probabilistically, are modeled by
continuous random variables. A number of important continuous distributions
are introduced in this chapter and, as in Chapter 6, we are also concerned with
the nature and applications of these distributions in science and engineering.

7.1 UNIFORM DISTRIBUTION

A continuous random variable X has a uniform distribution over an interval a to
b(b > a) if it is equally likely to take on any value in this interval. The probability
density function (pdf) of X is constant over interval (a, b) and has the form

8
1
< ; for a x b;
f x†ˆ b a 7:1†
X
0; elsewhere:
:
As we see from Figure 7.1(a), it is constant over (a, b), and the height must be
1/(b a) in order that the area under the density function is unity.
The probability distribution function (PDF) is, on integrating Equation (7.1),

0; for x < a;
8
>
>
x a
<
F X x†ˆ ; for a x b; 7:2†
b a
>
>
1; for x > b;
:

Fundamentals of Probability and Statistics for EngineersT. T. Soong  2004 John Wiley & Sons, Ltd
ISBNs: 0-470-86813-9 (HB) 0-470-86814-7 (PB)
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