Poisson Distribution curve is one of the most used statistical methods used for predicting event occurrence probability based on historical data of said event. Corporations use the Poisson distribution to predict sales for a financial quarter based on sales of the previous three quarters. Meteorologists use historical data to predict weather patterns for the upcoming year.

## Probability Mass Function

The probability mass function of the distribution is given by the following formula

In certain textbooks and references you will notice that the poisson PMF has different notations. x may be replaced by k and u may be replaced by λ (lambda)

The graph of the PMF for different k and λ values is given below

## Binomial vs Poisson

A new student of statistics may get confused when given a problem, in choosing between Binomial Distribution or Poisson. A simple way to differentiate between them is given below.

Binomial | Poisson |

You are calculating probability of an event occurring “x” times out of “n” trials | You are calculating number of events occurring within a specified time period |

You are given probability of event occurring and not occurring | You are given historical data of number of events occurred in previous time periods. |

## Poisson Probability Table

Some common Probability values of the poisson distribution are given below.

The Lambda (or u ) values are given in top row and expected occurrence values k (or x) are in left column.

This table is the numerical version of the graph shown above. A new statistics student will be able to quickly reference the graph and table to get the answer to common poisson probability problems.

You can learn more about poisson distribution here.