Poisson random variable

Situations occur where an event happens at random over a period of time:

We have to take a period of time where the rate is about unchanged. (not like the police calls in early morning/late afternoon).

Definition

A discrete random variable taking on values  k=0,1,2,... with the probability mass function:

is called the Poisson distribution.

We can check it is a probability mass function because

\begin{displaymath}\sum_{k=0}^{\infty} \frac{e^{-\lambda}\lambda^k}{k!}= e^{-\lambda}e^{\lambda}=1\end{displaymath}

Examples:
  1. Let X be the number of winning tickets among the PA lottery tickets sold in State College during a week.  Then,  calling winning tickets successes, we have that X is a binomial random variable. Since n, the total number of tickets sold in State College, is large, p, the probability that a ticket wins, is small, and the average number of winning tickets is appreciable, X is approximately a Poisson random variable.

  2. Let X be the number of misprints on a document page typed by a secretary.  Then X  is a binomial random variable if a word is called a success, provided that it is misprinted!  Since misprints are rare events, the number of words is large, and np, the average number of misprints, is of moderate value, X is approximately a Possion random variable.