Fundamentals of Probability and Statistics for Engineers

3.2.3 PROBABILITY DENSITY FUNCTION FOR CONTINUOUS

RANDOM VARIABLES

For a continuous random variable ,its PDF, is a continuous function
of and the derivative

exists for all The function is called the (pdf),
or simply the , of .(1)
Since is monotone nondecreasing, we clearly have

0 for all

Additional properties of can be derived easily from Equation (3.10);
these include

and

An example of pdfs has the shape shown in Figure 3.5. As indicated by
Equations (3.13), the total area under the curve is unity and the shaded area
from to gives the probability We again observe that the
knowledge of either pdf or PDF completely characterizes a continuous random
variable. The pdf does not exist for a discrete random variable since its
associated PDF has discrete jumps and is not differentiable at these points of
discontinuity.
Using the mass distribution analogy, the pdf of a continuous random variable
plays exactly the same role as the pmf of a discrete random variable. The

(^1) Note the use of upper-case and lower-case letters, PDF and pdf, to represent the distribution and
density functions, respectively.
44 Fundamentals of Probability and Statistics for Engineers
X FX 9 x),
x,
fX…x†ˆ
dFX…x†
dx

; … 3 : 10 †

x fXx probability density function

density function X

. 9 )

FX 9 x)

fX…x† x: … 3 : 11 †

fX 9 x)

FX…x†ˆ

Zx

1

fX…u†du; … 3 : 12 †

Z 1

1

fX…x†dxˆ 1 ;

P…a<Xb†ˆFX…b†FX…a†ˆ

Zb

a

fX…x†dx:

9

>>

=

>>

;

… 3 : 13 †

a b P 9 a<Xb).