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

hypergometric binomial normal
distribution n distribution np 5 distribution
0.1
N n11 p2 5
Figure 4-21 Conditions for approximating hypergeometric and binomial probabilities.

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 n N 0.1 . Recall that for a hypergeometric distribution p is defined as p K N. That is,
p is interpreted as the number of successes in the population. Therefore, the normal distribu-
tion 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 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.

Normal
Approximation to If X is a Poisson random variable with E1X2 and V1X2 ,
the Poisson
Distribution X
Z (4-13)
2

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

950 e 1000 1000
x
P1X 9502 a
x 0 x!
The computational difficulty is clear. The probability can be approximated as

950 1000
P1X x2 P aZ b P1Z 1.582 0.057
11000

EXERCISES FOR SECTION 4-7
4-61. Suppose that X is a binomial random variable with 4-62. Suppose that X is a binomial random variable with
n 200 and p 0.4. n 100 and p 0.1.
(a) Approximate the probability that X is less than or equal (a) Compute the exact probability that X is less than 4.
to 70. (b) Approximate the probability that X is less than 4 and com-
(b) Approximate the probability that X is greater than 70 and pare to the result in part (a).
less than 90. (c) Approximate the probability that 8 X 12 .