B. P. Lathi, Zhi Ding - Modern Digital and Analog Communication Systems-Oxford University Press (2009)

414 FUNDAMENTALS OF PROBABILITY THEORY
Properties of the CDF [Eqs. (8.25) and (8.26)] derived earlier are general and are valid
for continuous as well as discrete RVs.
Probability Density Function: From Eq. (8.26), we have
F x (x + x) = F x (x) + P(x < x :S x + x)
(8.27a)
If x --+ 0, then we can also express F x (x + x) via Taylor expansion as
dF x (x)
F x (X + x) '.::::'. F x (x) + -- x
dx
(8.27b)
From Eqs. (8.27), it follows that as x --+ 0,
dF x (X)
x = P(x < x :S x + x)
(8.28)
We designated the derivative of F x (x) with respect to x by Px(x) (Fig. 8.9),
dF x (x)
-- =px (x)
dx
(8.29)
The function P x (x) is called the probability density function (PDF) of the RV x. It follows
from Eq. (8.28) that the probability of observing the RV x in the interval (x, x + x) is
P x (x) x ( x --+ 0). This is the area under the PDF P x (x) over the interval x. as shown in
Fig. 8.9b.
Figure 8.9
(a) Cumulative
distribution
function (CDF).
(b) Probability
density function
(PDF).
Fx(x)
1 ---- ---::..;-----
---1
I
I
[ } Fx(x2) - Fx(x1 )
I
I
0
(a)
x------
0 X X1 X2
(b)
x -