4-7 NORMAL APPROXIMATION TO THE BINOMIAL AND POISSON DISTRIBUTIONS 119
EXAMPLE 4-17 In a digital communication channel, assume that the number of bits received in error can be
modeled by a binomial random variable, and assume that the probability that a bit is received
in error is. If 16 million bits are transmitted, what is the probability that more than
150 errors occur?
Let the random variable Xdenote the number of errors. Then Xis a binomial random vari-
able and
Clearly, the probability in Example 4-17 is difficult to compute. Fortunately, the normal
distribution can be used to provide an excellent approximation in this example.
P 1 X 1502 1 P 1 x 1502 1 a
150
x 0
a
16,000,000
x
b 110 ^52 x 11 10 ^52 16,000,000x
1 10 ^5
Figure 4-19 Normal
approximation to the
binomial distribution.
012345678910
0.00
0.05
0.10
0.20
0.25
x
f(x
)
n = 10
0.15
p = 0.5
If Xis a binomial random variable,
(4-12)
is approximately a standard normal random variable. The approximation is good for
np 5 and n 11 p 2 5
Z
Xnp
1 np 11 p 2
Normal
Approximation to
the Binomial
Distribution
Recall that for a binomial variable X, E(X) npand V(X)np(1 p). Consequently, the ex-
pression in Equation 4-12 is nothing more than the formula for standardizing the random vari-
able X. Probabilities involving Xcan be approximated by using a standard normal distribution.
The approximation is good when nis large relative to p.
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