The Poisson Probability Distribution

The Poisson probability distribution is an important discrete probability distribution for a number of applications, including:
1.    the number of failures in a large computer system during a given day.
2.    The number of replacement orders for a part received by a firm in a given month.
3.    The number of dents, scratches, or other defects in a large roll of sheet metal used to manufacture filters.

Assumptions of the Poisson Probability Distribution

Assume that an interval is divided into a very large number of subintervals so that the probability of the occurrence of an event in any subinterval is very small. The assumptions of a Poisson probability distribution are as follows:
1.    The probability of the occurrence of an event is constant for all subintervals.
2.    There can be no more than one occurrence in each subinterval.
3.    Occurrences are independent; that is, the occurrences in nonoverlapping intervals are independent of one another. 
      
We can derive the equation for computing Poisson probabilities directly from the binomial probability distribution by taking the mathematical limits as. With these limits the parameter is a constant that specifies the average number of occurrences (success) for a particular time and/or space.