Introductory Biostatistics

sciences have been documented. For example, the number of viruses in a solu-
tion, the number of defective teeth per person, the number of focal lesions in
virology, the number of victims of specific diseases, the number of cancer
deaths per household, and the number of infant deaths in a certain locality
during a given year, among others.
The probability density function of a Poisson distribution is given by

PrðX¼xÞ¼

yxe
y

x!

forx¼ 0 ; 1 ; 2 ;...

¼pðx;yÞ

The mean and variance of the Poisson distributionPðyÞare

m¼y

s^2 ¼y

However, the binomial distribution and the Poisson distribution are not totally
unrelated. In fact, it can be shown that a binomial distribution with largenand
smallpcan be approximated by the Poisson distribution withy¼np.
Given a sample of counts from the Poisson distributionPðyÞ,fxigin¼ 1 , the
sample meanxis an unbiased estimator fory; its standard error is given by

SEðxÞ¼

ffiffiffi

x

n

r

Example 10.1 In estimating the infection rates in populations of organisms,
sometimes it is impossible to assay each organism individually. Instead, the
organisms are randomly divided into a number of pools and each pool is tested
as a unit. Let

N¼number of insects in the sample

n¼number of pools used in the experiment

m¼number of insects per pool;N¼nmðfor simplicity;

assume thatmis the same for every poolÞ

The random variableXconcerned is the number of pools that show negative
test results (i.e., none of the insects are infected).
Letlbe the population infection rate; the probability that allminsects in a
pool are negative (in order to have a negative pool) is given by

p¼ð 1 
lÞm

Designating a negative pool as ‘‘success,’’ we have a binomial distribution for

POISSON DISTRIBUTION 351