5-3 Poisson Probability Distributions 233 ■ Number of Atlantic hurricanes in one year ■ Number of speling errors on a page ■ Number of machine failures in a month FORMULA 5-9 Poisson Probability Distribution P1x2 = mx # e-m x! where e ≈ 2.71828 m = mean number of occurrences of the event in the intervals Requirements for the Poisson Probability Distribution 1. The random variable x is the number of occurrences of an event in some interval. 2. The occurrences must be random. 3. The occurrences must be independent of each other. 4. The occurrences must be uniformly distributed over the interval being used. Parameters of the Poisson Probability Distribution • The mean is m. • The standard deviation is s = 1m. DEFINITION A Poisson probability distribution is a discrete probability distribution that applies to occurrences of some event over a specified interval. The random variable x is the number of occurrences of the event in an interval. The interval can be time, distance, area, volume, or some similar unit. The probability of the event occurring x times over an interval is given by Formula 5-9. Go Figure 53: Number of years before we run out of oil. Properties of the Poisson Probability Distribution 1. A particular Poisson distribution is determined only by the mean μ. 2. A Poisson distribution has possible x values of 0, 1, 2, . . . with no upper limit. Atlantic Hurricanes EXAMPLE 1 There have been 652 Atlantic hurricanes during the 118-year period starting in 1900. Assume that the Poisson distribution is a suitable model. a. Find m, the mean number of hurricanes per year. b. Find the probability that in a randomly selected year, there are exactly 6 hurricanes. That is, find P162, where P1x2 is the probability of x Atlantic hurricanes in a year. continued
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