Poisson Distribution
A Poisson distribution is the probability distribution that results from a Poisson
experiment.
Attributes of a Poisson Experiment
A Poisson experiment is a
statistical experiment that has the following properties:

The probability that a success will occur in an extremely small region is
virtually zero.
Note that the specified region could take many forms. For instance, it could be
a length, an area, a volume, a period of time, etc.
Notation
The following notation is helpful, when we talk about the Poisson distribution.

P(x; μ): The Poisson probability that exactly x
successes occur in a Poisson experiment, when the mean number of
successes is μ.
Poisson Distribution
A Poisson random variable is the number of successes that
result from a Poisson experiment. The
probability distribution of a Poisson random variable is called a Poisson
distribution.
Given the mean number of successes (μ) that occur in a specified region,
we can compute the Poisson probability based on the following formula:
Poisson Formula. Suppose we conduct a
Poisson experiment, in which the average number of successes within a given
region is μ. Then, the Poisson probability is:
P(x; μ) = (e^{μ}) (μ^{x}) / x!
where
x is the actual number of successes that result from the
experiment, and
e is approximately equal to 2.71828.
The Poisson distribution has the following properties:
Example 1
The average number of homes sold by the Acme Realty company is 2 homes per day.
What is the probability that exactly 3 homes will be sold tomorrow?
Solution: This is a Poisson experiment in which we know the following:

e = 2.71828; since e is a constant equal to approximately 2.71828.
We plug these values into the Poisson formula as follows:
P(x; μ) = (e^{μ}) (μ^{x}) / x!
P(3; 2) = (2.71828^{2}) (2^{3}) / 3!
P(3; 2) = (0.13534) (8) / 6
P(3; 2) = 0.180
Thus, the probability of selling 3 homes tomorrow is 0.180 .
Poisson
Calculator
Clearly, the Poisson formula requires many timeconsuming computations. The Stat
Trek Poisson Calculator can do this work for you  quickly, easily, and
errorfree. Use the Poisson Calculator to compute Poisson probabilities and
cumulative Poisson probabilities. The
calculator is free. It can be found under the Stat Tables
tab, which appears in the header of every Stat Trek web page.
Cumulative Poisson Probability
A cumulative Poisson probability refers to the probability that
the Poisson random variable is greater than some specified lower limit
and less than some specified upper limit.
Example 1
Suppose the average number of lions seen on a 1day safari is 5. What is the
probability that tourists will see fewer than four lions on the next 1day
safari?
Solution: This is a Poisson experiment in which we know the following:

x = 0, 1, 2, or 3; since we want to find the likelihood that tourists will see
fewer than 4 lions; that is, we want the probability that they will see 0, 1,
2, or 3 lions.

e = 2.71828; since e is a constant equal to approximately 2.71828.
To solve this problem, we need to find the probability that tourists will see 0,
1, 2, or 3 lions. Thus, we need to calculate the sum of four probabilities:
P(0; 5) + P(1; 5) + P(2; 5) + P(3; 5). To compute this sum, we use the Poisson
formula:
P(x < 3, 5) = P(0; 5) + P(1; 5) + P(2; 5) + P(3; 5)
P(x < 3, 5) = [ (e^{5})(5^{0}) / 0! ] + [ (e^{5})(5^{1})
/ 1! ] + [ (e^{5})(5^{2}) / 2! ] + [ (e^{5})(5^{3})
/ 3! ]
P(x < 3, 5) = [ (0.006738)(1) / 1 ] + [ (0.006738)(5) / 1 ] + [
(0.006738)(25) / 2 ] + [ (0.006738)(125) / 6 ]
P(x < 3, 5) = [ 0.0067 ] + [ 0.03369 ] + [ 0.084224 ] + [ 0.140375 ]
P(x < 3, 5) = 0.2650
Thus, the probability of seeing at no more than 3 lions is 0.2650.