Introductory Biostatistics

3.4.2 Poisson Distribution

The next discrete distribution that we consider is the Poisson distribution,
named after a French mathematician. This distribution has been used exten-
sively in health science to model the distribution of the number of occurrences
xof some random event in an interval of time or space, or some volume of
matter. For example, a hospital administrator has been studying daily emer-
gency admissions over a period of several months and has found that admis-
sions have averaged three per day. He or she is then interested in finding the
probability that no emergency admissions will occur on a particular day. The
Poisson distributionis characterized by its probability density function:

PrðX¼xÞ¼

yxe
y

x!

forx¼ 0 ; 1 ; 2 ;...

It turns out, interestingly enough, that for a Poisson distribution the variance is
equal to the mean, the parameteryabove. Therefore, we can answer probabil-
ity questions by using the formula for the Poisson density above or by con-
verting the number of occurrencesxto the standard normal score, provided
thatyb10:

z¼

x
y

ffiffiffi

y

p

In other words, we can approximate a Poisson distribution by a normal distri-
bution with meanyifyis at least 10.
Here is another example involving the Poisson distribution. The infant
mortality rate (IMR) is defined as

IMR¼

d

N

for a certain target population during a given year, wheredis the number of
deaths during the first year of life andNis the total number of live births. In
the studies of IMRs,Nis conventionally assumed to be fixed anddto follow a
Poisson distribution.

Example 3.9 For the year 1981 we have the following data for the New
England states (Connecticut, Maine, Massachusetts, New Hampshire, Rhode
Island, and Vermont):

d¼ 1585

N¼ 164 ; 200

For the same year, the national infant mortality rate was 11.9 (per 1000 live

136 PROBABILITY AND PROBABILITY MODELS