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

8.7 Central Limit Theorem 447
is known as the sample mean. The interpretation is that the sample mean of any distribution
with nonzero finite variance converges to Gaussian distribution with fixed mean µ, and
decreasing variance a 2 /n. In other words, regardless of the true distribution ofxi, L7= i Xi
can be approximated by a Gaussian distribution with mean nµ, and variance na 2 .
Example 8.25 Consider a communication system that transmits a data packet of 1024 bits. Each bit can be
in error with probability of 10 - 2 . Find the (approximate) probability that more than 30 of the
1024 bits are in error.
Define a random variable Xi such that Xi = 1 if the ith bit is in error and x; = 0 if not.
Hence
1024
V = LXj
i=l
is the number of errors in the data packet. We would like to find P(v > 30).
Since P(xi = 1) = 10 - 2 and P(xi = 0) = 1 - 10 - 2 , strictly speaking we would
need to find
1024
P(v > 30) = L
m
m=31
( 1024 )
m 1024-m
(10 - 2 ) (1 - 10 - 2 )
This calculation is time-consuming. We now apply the central limit theorem to solve this
problem approximately.
First, we find
As a result,
Xi = 10 -2 X (1) + (1 - 10 -2 ) X (0) = 10 -2
x? ' = 10 -2 X (1) 2 + (1 - 10 -2 ) X (0) = 10 -2
a? = x; - (xii = 0.0099
Based on the central limit theorem, v = L; x; is approximately Gaussian with mean of
1024 • 10 -2 = 10.24 and variance 1024 x 0.0099 = 10.1 376. Since
V - 10.24
y =
✓10.1 376
is a standard Gaussian with zero mean and unit variance,
30 - 10.24
P(v > 30) = P ( y > ===- )
✓10.1376
= P(y > 6.20611)
= Q(6.2061 l)
'.:::'. 1.925 X 10 - lO