2 within a single interval an infinite number of

2. Within a single interval, an infinite number of
occurrences of the event are theoretically possible.
We are not restricted to a fixed number of trials.
3. The events occur independently both within the same
interval and between consecutive intervals.

91
Example: Suppose X represents the number of individuals
involved in a motor vehicle accident each year and X ~ Poisson
with

= 2.4.
a. P(X = 0) =
b. P(X = 1) =
c. P(X = 2) =
d. P(X = 3) =
e. P(X = 4) =
f. P(X = 5) =
g. P(X > 5) =

92
Center and Spread of the Poisson Distribution
The mean and the variance of the Poisson distribution are
identical and can be represented by the single parameter
 .

93
Example: The number of cases of tetanus reported in the United
States during a single month in 1989 has a Poisson distribution
with parameter

= 4.5.
a. What is the probability that exactly one case of tetanus will be
reported?
b. What is the probability that at most two cases of tetanus will
be reported?
c. What is the probability that four or more cases will be
reported?
d. What is the mean number of cases of tetanus reported in a
one-month period? What is the standard deviation?

94
Poisson Approximation to the Binomial
The binomial distribution with large n and small p can be
accurately approximated by a Poisson distribution with
parameter  = np.
A rule of thumb is to use the approximation when n  100
and p  .01.

95
Example: Let X be a random variable that represents the number
of infants in a group of 500 who die before reaching their first
birthdays. In the U.S., the probability that a child dies during his
or her first year of life is 0.0085.
a. What is the mean number of infants who would die in a group
of this size?
b. What is the probability that at most five infants out of 500 die
in their first year of life?
c. What is the probability that between 5 and 10 infants inclusive
die in their first year of life?