26 Jun 2001

### The Poisson Distribution

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 lx ��� 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.

### Expectation & Variance

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

### The Recurrence Formula

 P(X = x + 1) ������ P(X = x) �=� l ��� (x + 1)

### Two Independent Poisson Variables

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

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