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3-9 POISSON DISTRIBUTION 89

sharpness. If any dull blade is found, the assembly is replaced (c) What is the probability that 4 of the 6 numbers chosen by
with a newly sharpened set of blades. a player appear in the state’s sample?
(a) If 10 of the blades in an assembly are dull, what is the (d) If a player enters one lottery each week, what is the
probability that the assembly is replaced the first day it is expected number of weeks until a player matches all 6
evaluated? numbers in the state’s sample?
(b) If 10 of the blades in an assembly are dull, what is the 3-95. Continuation of Exercises 3-86 and 3-87.
probability that the assembly is not replaced until the third (a) Calculate the finite population corrections for Exercises
day of evaluation? [Hint: Assume the daily decisions are 3-86 and 3-87. For which exercise should the binomial
independent, and use the geometric distribution.] approximation to the distribution of X be better?
(c) Suppose on the first day of evaluation, two of the blades (b) For Exercise 3-86, calculate P1X 12 and P1X 42 as-
are dull, on the second day of evaluation six are dull, and suming that X has a binomial distribution and compare
on the third day of evaluation, ten are dull. What is the these results to results derived from the hypergeometric
probability that the assembly is not replaced until the third distribution.
day of evaluation? [Hint: Assume the daily decisions are (c) For Exercise 3-87, calculate P1X 12 and P1X 42
independent. However, the probability of replacement assuming that X has a binomial distribution and compare
changes every day.] these results to the results derived from the hypergeometric
3-94. A state runs a lottery in which 6 numbers are ran- distribution.
domly selected from 40, without replacement. A player 3-96. Use the binomial approximation to the hypergeo-
chooses 6 numbers before the state’s sample is selected. metric distribution to approximate the probabilities in
(a) What is the probability that the 6 numbers chosen by a Exercise 3-92. What is the finite population correction in this
player match all 6 numbers in the state’s sample? exercise?
(b) What is the probability that 5 of the 6 numbers chosen by
a player appear in the state’s sample?

3-9 POISSON DISTRIBUTION

We introduce the Poisson distribution with an example.

EXAMPLE 3-30 Consider the transmission of n bits over a digital communication channel. Let the random
variable X equal the number of bits in error. When the probability that a bit is in error is con-
stant and the transmissions are independent, X has a binomial distribution. Let p denote the
probability that a bit is in error. Let pn . Then, E1x2 pn and

n x n x n x n x
P1X x2 a b P 11 p2 a b a b a1 b
x x n n
Now, suppose that the number of bits transmitted increases and the probability of an error
decreases exactly enough that pn remains equal to a constant. That is, n increases and p de-
creases accordingly, such that E(X) remains constant. Then, with some work, it can be
shown that

e x
P1X x2
lim nS , x 0, 1, 2, p
x!
Also, because the number of bits transmitted tends to infinity, the number of errors can equal
any nonnegative integer. Therefore, the range of X is the integers from zero to infinity.

The distribution obtained as the limit in the above example is more useful than the deri-
vation above implies. The following example illustrates the broader applicability.