4-7 NORMAL APPROXIMATION TO THE BINOMIAL AND POISSON DISTRIBUTIONS

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We began our section on the normal distribution with the central limit theorem and the normal distribution as an approximation to a random variable with a large number of trials. Consequently, it should not be a surprise to learn that the normal distribution can be used to approximate binomial probabilities for cases in which n is large. The following example illustrates that for many physical systems the binomial model is appropriate with an extremely large value for n. In these cases, it is dif cult to calculate probabilities by using the binomial distribution. Fortunately, the normal approximation is most effective in these cases. An illustration is provided in Fig. 4-19. The area of each bar equals the binomial probability of x. Notice that the area of bars can be approximated by areas under the normal density function.

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4-7 NORMAL APPROXIMATION TO THE BINOMIAL AND POISSON DISTRIBUTIONS

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0.25 n = 10 p = 0.5 0.20

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0.15 f(x) 0.10 0.05

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Figure 4-19 Normal approximation to the binomial distribution.

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0.00 0 1 2 3 4 5 x 6 7 8 9 10

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EXAMPLE 4-17

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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 1 10 5. If 16 million bits are transmitted, what is the probability that more than 150 errors occur Let the random variable X denote the number of errors. Then X is a binomial random variable and P 1X 1502 1 P1x 1502 1 aa

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16,000,000 b 110 5 2 x 11 x

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10 5 2 16,000,000

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Clearly, the probability in Example 4-17 is dif cult to compute. Fortunately, the normal distribution can be used to provide an excellent approximation in this example.

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Normal Approximation to the Binomial Distribution

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If X is a binomial random variable, Z

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X np 1np11 p2

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(4-12)

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is approximately a standard normal random variable. The approximation is good for np 5 and n11 p2 5

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Recall that for a binomial variable X, E(X) np and V(X) np(1 p). Consequently, the expression in Equation 4-12 is nothing more than the formula for standardizing the random variable X. Probabilities involving X can be approximated by using a standard normal distribution. The approximation is good when n is large relative to p.

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CHAPTER 4 CONTINUOUS RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS

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EXAMPLE 4-18

The digital communication problem in the previous example is solved as follows: P1X 1502 Pa P1Z Because np 116 106 211 10 5 2 is expected to work well in this case. X 160 10 2

216011 150

160 0.785

10 5 2

160 and n(1

p) is much larger, the approximation

EXAMPLE 4-19

Again consider the transmission of bits in Example 4-18. To judge how well the normal approximation works, assume only n 50 bits are to be transmitted and that the probability of an error is p 0.1. The exact probability that 2 or less errors occur is P1X 22 a 50 b 0.950 0 50 a b 0.110.949 2 1 50 a 2 b 0.12 10.948 2 0.112

Based on the normal approximation P1X 22 Pa X 5 2.12 2 5 b 2.12 P1Z 1.422 0.08

Even for a sample as small as 50 bits, the normal approximation is reasonable. If np or n(1 p) is small, the binomial distribution is quite skewed and the symmetric normal distribution is not a good approximation. Two cases are illustrated in Fig. 4-20. However, a correction factor can be used that will further improve the approximation. This factor is called a continuity correction and it is discussed in Section 4-8 on the CD.

0.4 n p 10 0.1 10 0.9 0.3

f(x)

Figure 4-20 Binomial distribution is not symmetrical if p is near 0 or 1.

0.0 0 1 2 3 4 5 x 6 7 8 9 10

4-7 NORMAL APPROXIMATION TO THE BIOMIAL AND POISSON DISTRIBUTIONS

hypergometric distribution

Figure 4-21

binomial distribution

np 5 n11 p2 5

normal distribution

Conditions for approximating hypergeometric and binomial probabilities.

Recall that the binomial distribution is a satisfactory approximation to the hypergeometric distribution when n, the sample size, is small relative to N, the size of the population from which the sample is selected. A rule of thumb is that the binomial approximation is effective if n N 0.1. Recall that for a hypergeometric distribution p is de ned as p K N. That is, p is interpreted as the number of successes in the population. Therefore, the normal distribution can provide an effective approximation of hypergeometric probabilities when n N 0.1, np 5 and n(1 p) 5. Figure 4-21 provides a summary of these guidelines. Recall that the Poisson distribution was developed as the limit of a binomial distribution as the number of trials increased to in nity. Consequently, it should not be surprising to nd that the normal distribution can also be used to approximate probabilities of a Poisson random variable.