# Poisson Distribution Calculator

Sometimes you may have an idea of the expected number of successes in a range or interval. In insurance, it is common to collect data that indicates the expected number of accidents a cohort of drivers may have in a year. This situation can be modelled using a Poisson distribution. Questions can be answered by this distribution such as what is the probability that a cohort will have 300 accidents this year given the assumed expected number of auto accidents in one year is 400. Or one could ask what is the probability of having 450 or more accidents in a given year if you believe the expected number of accidents in a year to be 500.

The probability of observing x occurrences over the specified interval (whether it be time, length, etc.) is represented by the probability density function (pdf) P(X = x) = e-n * nx / x!, where n is equal to the expected number of successes in a range or interval.

The mean of such a distribution is n and the variance is also n.

You can use the calculator below to calculate the probabilities under your own scenario using the assumptions you input. The probability density function yields probabilities for a number of intervals or ranges, whereas the cumulative density function yields probabilities for each and every number of intervals or ranges up to a specified number. The graphs generated are shaped in accordance with the assumptions you enter.