1. The Poisson probability distribution

2. Definition of distribution
A discrete random variable X is said to have a Poisson distribution with parameter if the pmf of X is
for x satisfying

3. Mean and variance of a Poisson random variable
If X has Poisson distribution with parameter , then

4. Is this a legitimate probability distribution?
The assumption that ensures that .
That the probabilities sum to 1 is a consequence of the Maclaurin series for :

5. The Poisson distribution as a limit
Suppose that in the binomial pmf b(x;n,p) we let
and in such a way that np approaches a value Then

6. Meaning of proposition of Poisson limit
According to this proposition, in any binomial experiment in which n is large and p is small, .
As a rule of thumb, this approximation can safely be applied if n > 50 and np < 5.

7. Example
If a publisher of nontechnical books takes great pains to ensure that its books are free of typographical errors, so that the probability of any given page containing at least one such error is .005 and errors are independent from page to page, what is the probability that one of its 400-page novels will contain exactly one page with errors?

8. Solution to the example
Let S be a page with at least one error, let F be a page with no errors, and let X be the number of pages with at least one error.
Then X is binomial with n = 400, p = .005, and np=2.

9. Approximate solution to the example
The Poisson approximation is ,
which is very close to the true answer.
We used in the approximation.

10. Poisson process
An important application of the Poisson distribution arises in connection with the occurrence of events over time.
The events might be visits to a website, messages to a particular address, or accidents in an industrial facility.

11. Assumptions for a Poisson process
There exists a parameter such that for any short interval of length , the probability that exactly one event occurs is
The probability of more than one event occurring during is
The number of events occurring during the time interval is independent of the number that occur prior to this interval.

12. Proposition
Under the assumptions above, the probability of k events in a time interval of length t is , i.e. Poisson with parameter
Thus the expected number of events in an interval of length t is , and the expected number in a unit interval is .

13. Example
Suppose pulses arrive at a counter at an average rate of six per minute, so that
To find the probability that at least one pulse is received in a ½-minute interval we use the Poisson distribution with parameter , and thus