A drv X is said to have a Poisson distribution
with parameter l (> 0) if its pdf is given by

P(X = x) = e^{-l}

l^{x}

���

x!

,�� for x = 0, 1, 2, ... .

We write X ~ Po(l).

The Poisson distribution was introduced by S. D. Poisson.

Note that there is no upper limit to the value of x.

Uses of the Poisson Distribution

There are two main uses of the Poisson Distribution.

1.��� It is frequently used when considering the distribution
of events which occur randomly in time or
space.

If an event is randomly scattered in space or time, and has mean number
of occurence l in a given interval of space
or time, and if X is the rv 'the number of occurences in the given
interval', then X ~ Po(l).

Examples of random variables that usually follows a Poisson distribution:

The number of misprints in a page of a book.

The number of customers entering a bank on a given
day.

The number of bacteria in a certain amount of liquid.

The number of a-particles
discharged in a given period of time from some radioactive material.

The number of phone calls to a company on a given
day.

2.��� It is used as an approximation
to the binomial distribution under situable conditions.

Suppose X ~ B(n, p).� It can be shown that
if n is large (say n � 50),
p is small (say p � 0.1),
and np � 5, then X can be
approximated by the Poisson distribution with l
= np, ie, X ~ Po(np) approx.

An Excel 97 worksheet to show
the approxmiation.� Change only the values of n and p.