Poisson Distribution: Calculation of Probabilities

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Many studies are based on counts of the times a particular event occurs in a given area of opportunity. An area of opportunity is a continuous unit or interval of time, volume, or any physical area in which there can be more than one occurrence of an event. Examples of variables that follow the Poisson distribution are the surface defects on a new refrigerator, the number of network failures in a day, the number of people arriving at a bank, and the number of fleas on the body of a dog. You can use the Poisson distribution to calculate probabilities in situations such as these if the following properties hold:

You are interested in counting the number of times a particular event occurs in a given area of opportunity. The area of opportunity is defined by time, length, surface area, and so forth. •The probability that an event occurs in a given area of opportunity is the same for all the areas of opportunity. •The number of events that occur in one area of opportunity is independent of the number of events that occur in any other area of opportunity. •The probability that two or more events will occur in an area of opportunity approaches zero as the area of opportunity becomes smaller.

Consider the number of customers arriving during the lunch hour at a bank located in the central business district in a large city. You are interested in the number of customers who arrive each minute. Does this situation match the four properties of the Poisson distribution given earlier? First, the event of interest is a customer arriving, and the given area of opportunity is defined as a one-minute interval. Will zero customers arrive, one customer arrives, and two customers arrive, and so on? Second, it is reasonable to assume that the probability that a customer arrives during a particular one-minute interval is the same as the probability for all the other one minute intervals. Third, the arrival of one customer in any one-minute interval has...
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