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Poisson Probability Model

Another discrete probability model of interest is the Poisson . Actually an understanding of the Poisson will help you get through the frustration of queues. You know, "When I came into this store there was no one in the line (queue) at the fast checkout, but when I went to check out there were 6 people ahead of me."

Poisson random variables deal with counts of events over time.

1.
The number of customers entering a deli over, say, noon hour.
2.
The number of calls entering a switch board from 9 to 10 in the morning.
3.
The number of tornadoes which touchdown in Kalamazoo County in July.
4.
The number of plane crashes in one year.
These are all examples of random variables which are counts of events over time.

There are two axioms for a Poisson process.

1.
In a small interval of time, the probability of an event occurring is $\lambda$  times the length of the interval of time. Further, the probability that two or more events occur is practically 0.
2.
If two intervals of time do not overlap, the occurrence or non occurrence of events in these intervals are independent of one another.
Consider a bank. It seems reasonable, that two customers will generally not enter the bank simultaneously, (i.e., in a short interval of time at most one). And the longer the short interval of time, the more probable it becomes that some customer will enter. Further the customers that enter between 9 and 10 are independent of those entering between 10 and 11.

These are assumptions and for a given situation they may or may not be true. For example, if a motorist is driving to a drive-in bank window and he sees a long queue then he may decide to do his banking later; i.e., dependence broke down. Actually these assumptions never truly hold but often the approximation is close to reality and predictions based on the model are often fairly accurate.

Let X denote the number of events in one unit of time. Then under these assumptions we can obtain the probability model for X. The range of X is {0, 1, 2, 3, ... } and its probability distribution can be obtained. The mean of X is $\lambda$ and its variance is also $\lambda$.

As with the binomial, we use the probability module to obtain the probabilities. To obtain the probability that X = k , the input consists of k and $\lambda$.


next up previous contents index
Next: Continuous Probability Models Up: Discrete Populations (Probability Models) Previous: Binomial Probability Model

2001-01-01