Applied Statistics and Probability for Engineers

4-7 NORMAL APPROXIMATION TO THE BIOMIAL AND POISSON DISTRIBUTIONS 121

Recall that the binomial distribution is a satisfactory approximation to the hypergeomet-

ric 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. Recall that for a hypergeometric distribution pis defined as That is,

pis interpreted as the number of successes in the population. Therefore, the normal distribu-

tion can provide an effective approximation of hypergeometric probabilities when nN 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 infinity. Consequently, it should not be surprising to find that the

normal distribution can also be used to approximate probabilities of a Poisson random variable.

n N0.1 pK N.

If Xis a Poisson random variable with and

(4-13)

is approximately a standard normal random variable. The approximation is good for

 5

Z

X

2 

E 1 X 2  V 1 X 2 ,

Normal

Approximation to

the Poisson

Distribution

hypergometric  binomial  normal

distribution n distribution np^5 distribution

N

0.1

Figure 4-21 Conditions for approximating hypergeometric and binomial probabilities.

n 11 p 2  5

EXAMPLE 4-20 Assume that the number of asbestos particles in a squared meter of dust on a surface follows

a Poisson distribution with a mean of 1000. If a squared meter of dust is analyzed, what is the

probability that less than 950 particles are found?

This probability can be expressed exactly as

The computational difficulty is clear. The probability can be approximated as

EXERCISES FOR SECTION 4-7

P 1 Xx 2 P aZ

950  1000

11000

bP 1 Z1.58 2 0.057

P 1 X 9502  a

950

x 0

e^1000 x^1000

x!

4-61. Suppose that Xis a binomial random variable with

and

(a) Approximate the probability that Xis less than or equal

to 70.

(b) Approximate the probability that Xis greater than 70 and

less than 90.

n 200 p0.4.

4-62. Suppose that Xis a binomial random variable with

n100 and p0.1.

(a) Compute the exact probability that Xis less than 4.

(b) Approximate the probability that Xis less than 4 and com-

pare to the result in part (a).

(c) Approximate the probability that 8 X 12.

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