Sums of Poisson Distributions

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

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

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

It is only a small step from using this scaling property of the Poisson distribution to finding the sum of two Poisson distributions.

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

If the accidents on each road are independent of each other, then for the first road we can model the number of accidents per month on the first road by a Poisson distribution with mean 2,and the number of accidents on the second road by a Poisson distribution with mean 3,

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

In this case then,

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

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

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