Poisson Distribution:

A discrete probability distribution that gives the probability of a given number of events k occurring in a fixed interval of time is known as Poisson distribution. The Poisson distribution is used to calculate the probabilities of number of successes based on the mean number of successes. In a Poisson distribution, events are assumed to occur with a known constant rate, µ, independent of the time since the last event.

P(X= x) =

Where e is the base of natural logarithm as in   (2.7183)
μ is the mean number of “successes”
x is the number of “successes”

The events occur in disjoint intervals (non-overlapping).

Two or more events cannot occur simultaneously.

Each event occurs at a constant rate.

Mean = μ.

Variance = μ.

Example:

The mean number of calls to a fire station on a weekday is 8. What is the probability that there would be 11 calls on a given weekday?

P= = 0.072

Let us solve this example in Excel using the function, POISSON(x,mean,cumulative)

X   – number of events (11).

Mean   – expected numeric value(8).

cumulative is TRUE, if the number of random events occurring will be between 0 and x inclusive; FALSE, if the number of events occurring will be exactly x.

Here for this example we need cumulative to be FALSE.