## Sums of Poisson Distributions

If a random variable has aPoisson distribution, so that in some sense, events occur at acertain rate, then we can scale the distribution, so that theinterval over which events occur changes.

If print errors occur at the rateof three per page, then they occur at the rate of one per third of apage or 12 per four pages.

If accidents occur at the rate oftwo per day then they occur at the rate of one accident per half dayor 14 accidents per week.

It is only a small step fromusing this scaling property of the Poisson distribution to findingthe sum of two Poisson distributions.

If accidents occur on two roadsat the rate of two per month on one road and three per month on theother, then accidents occur overall at the rate of five per month.

If the accidents on each road areindependent of each other, then for the first road we can model thenumber of accidents per month on the first road by a Poissondistribution with mean 2,andthe number of accidents on the second road by a Poisson distributionwith mean 3,

If the accidents on the firstroad are independent of accidents on the second road and vice versa,then all the accidents are independent of each other, and we canmodel the overall number of accidents per month by

In this case then,

In general, for events whichoccurs are some rate %lambda-1 and %lambda-2 per time period (thetime period must be the same), and each event is independent of eachother the can model the total number of events in the same timeperiod by of the sum of the two rates.

We can generalise this to expressthe sum of any number of Poisson distributions as a single Poissondistribution, as long as each rate is expressed in terms of the samettime perion (per week OR per month etc).