Page 108 - Applied Statistics And Probability For Engineers

PQ220 6234F.Ch 03 13/04/2002 03:19 PM Page 86

86 CHAPTER 3 DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS

Let X equal the number of parts in the sample from the local supplier. Then, X has a
hypergeometric distribution and the requested probability is P1X 42. Consequently,

100 200
a b a b
4 0
P1X 42 0.0119
300
a b
4
What is the probability that two or more parts in the sample are from the local supplier?

100 200 100 200 100 200
a b a b a b a b a b a b
2 2 3 1 4 0
P1X 22
300 300 300
a b a b a b
4 4 4
0.298 0.098 0.0119 0.408
What is the probability that at least one part in the sample is from the local supplier?

100 200
a b a b
0 4
P1X 12 1 P1X 02 1 0.804
300
a b
4
The mean and variance of a hypergeometric random variable can be determined from
the trials that comprise the experiment. However, the trials are not independent, and so the
calculations are more difficult than for a binomial distribution. The results are stated as
follows.

If X is a hypergeometric random variable with parameters N, K, and n, then

N n
2
E1X2 np and V1X2 np11 p2 a b (3-14)
N 1
where p K N .

Here p is interpreted as the proportion of successes in the set of N objects.

EXAMPLE 3-28 In the previous example, the sample size is 4. The random variable X is the number of parts in
the sample from the local supplier. Then, p 100 300 1 3 . Therefore,

E1X2 41100 3002 1.33