Applied Statistics and Probability for Engineers

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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