## Mean and Variance of the Poisson Distribution

The Poisson distribution models events occurring at a constant average rateso that in each fixed time period we expectevents to occur.

The probability thatevents will occur in a single fixed time period isand the variance of the number of events in each time period is also equal toThe equality of the mean and variance for the Poisson distribution can be used as evidence that a Poisson distribution can be used to model a phenomena.

We can derive the mena of a Poisson distribution from (1)

sinceby definition of a probability distribution

Similarly

Now we must perform some trickery. Writeso that the last summation above becomes

Then