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

412 FUNDAMENTALS OF PROBABILITY THEORY
Example 8.14 Over a certain binary communication channel, the symbol O is transmitted with probability 0.4
and 1 is transmitted with probability 0.6. It is given that P(E IO) = 10 -6 and P(E ll) = 10 - 4,
where P( E Ix;) is the probability of detecting the error given that Xi is transmitted. Determine
P(E), the error probability of the channel.
If P( E, x;) is the joint probability that Xi is transmitted and it is detected wrongly, then the
total probability theorem yields
P(E) = LP(E lx;)P(x;)
= Px (O)P(EIO) + Px (l)P(E ll)
= 0.4(10 -6 ) + 0.6(10 -4 )
= 0.604(10 -4 )
Note that P(E IO) = 10 -6 means that on the average, one out of 1 million received
Os will be detected erroneously. Similarly, P(E ll) = 10 -4 means that on the average, one
out of 10,000 received ls will be in error. But P(E) = 0.604(10 -4 ) indicates that on the
average, one out of 1/0.604(10 -4 ) 1 6,556 digits (regardless of whether they are ls
or Os) will be received in error.
Cumulative Distribution Function
The cumulative distribution function ( CDF) F x (x) of an RV x is the probability that x takes
a value less than or equal to x; that is,
F x (x) = P(x :S x)
(8.24)
We can show that a CDP F x (x) has the following four properties:
1. F x (x) 2:. 0
2. Fx (oo) = 1
3. Fx(-oo) = 0
4. F x (x) is a nondecreasing function, that is,
Fx (x1) S F x (x2) for x1 S x2
(8.25a)
(8.25b)
(8.25c)
(8.25d)
(8.25e)
The first property is obvious. The second and third properties are proved by observing that
Fx (oo) = P(x :S oo) and Fx (-oo) = P(x :S -oo). To prove the fourth property, we have,
from Eq. (8.24),
F x (x2) = P(x :S x2)
= P[(x :S x1) U (x1 < x :S x2)]