The difference between the two is that while both measure the number of certain random events (or "successes") within a certain frame, the Binomial is based on discrete events, while the Poisson is based on continuous events. That is, with a binomial distribution you have a certain number, nn, of "attempts," each of which has probability of success pp. With a Poisson distribution, you essentially have infinite attempts, with infinitesimal chance of success. That is, given a Binomial distribution with some n,pn,p, if you let n→∞n→∞ and p→0p→0 in such a way that np→λnp→λ, then that distribution approaches a Poisson distribution with parameter λλ.

Because of this limiting effect, Poisson distributions are used to model occurences of events that could happen a very large number of times, but happen rarely. That is, they are used in situations that would be more properly represented by a Binomial distribution with a very large nn and small pp, especially when the exact values of nn and ppare unknown. (Historically, the number of wrongful criminal convictions in a country)

If μ is the average number of successes occurring in a given time interval or region in the Poisson distribution, then the mean and the variance of the Poisson distribution are both equal to μ. Note: In a Poisson distribution, only one parameter, μ is needed to determine the probability of an event.

Poisson distribution

The horizontal axis is the index X. The function is defined only at integer values of X. The connecting lines are only guides for the eye and do not indicate continuity. Notice that as λ increases the distribution begins to resemble a normal distribution.