Poisson Distribution Help

What do these properties mean?

• Discrete Event

  The discrete events are what are being observed over the continuous interval. These events could be any outcome that is of interest to the observer and can be measured to be distinct from another event. Because the number of events observed in the interval is the random variable, it means that it is the thing in the Poisson process that is likely to change from process to process.

• Continuous Interval

  This means that the interval over which events are observed is measured continuously. For example, time, length, and space are all measured continuously, so any of them could be the interval over which the events are observed.

Example

Poisson processes are unique from the other discrete distributions in that they do not represent an experiment - rather, they identify a process. This makes them easy to identify compared to the other discrete distributions, because if you see or hear the word "process" you are dealing with a Poisson distribution.

For example, imagine that a you are observing the number of spam emails you receive over a period of one week. You know from previous observation that you expect to receive 6.5 spam emails each week, with a variance of 6.5. This would be a Poisson distribution with k = 6.5 (because the mean and variance of the number of spam emails you receive in a week is 6.5 emails). The only thing left would be to determine how many spam emails you would like to receive that week, and that would be x (because it is the number that you are hoping to observe over the interval). Note that you will not always receive that number of spam emails over the period of a week - there is only some probability that you will.

Requirements

• The mean and variance must be greater than zero (k > 0).
• The number of events observed in the set interval must be a whole number greater than or equal to zero (x = 0, 1, 2 ...)