Page 62 - Fundamentals of Probability and Statistics for Engineers

Random Variables and Probability Distributions 45

f (x)
X

x
a b
Figure 3.5 A probability density function, f 9x)
X
function f 9x) can be interpreted as the mass density (mass per unit length).
X
There are no masses attached to discrete points as in the discrete random
variable case. The use of the term density function is therefore appropriate here
for f 9x).
X
Example 3.3. A random variable X for which the density function has the
form 9a > 0):

ax
ae ; for x 0;
f x†ˆ 3:14†
X 0; elsewhere;
is said to be exponentially distributed. We can easily check that all the condi-
tions given by Equations (3.11)–(3.13) are satisfied. The pdf is presented
graphically in Figure 3.6(a), and the associated PDF is shown in Figure 3.6(b).
The functional form of the PDF as obtained from Equation (3.12) is

X
f (x) F (x)
X

1
a

x x
0 1 0

(a) (b)
Figure 3.6 (a) Probability density function, f 9x), and (b) probability distribution
X
function, F X 9x), for random variable X in Example 3.3

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