Poisson Distribution

26 Jun 2001

Main Points

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 ž 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.

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.

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.

n is large

p is small

np Ł 5

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

If X ~ Po(l), then E(X) = l, Var (X) = l.

P(X = x + 1) žžžžžž P(X = x)

l žžž (x + 1)

If X and Y are independent variables with X ~ Po(l₁) and Y ~ Po(l₂), then X + Y ~ Po(l₁ + l₂).