The Poisson distribution (Section 5.4) gives the probability for the number of occurrences for a "rare" event. Now, let x be a random variable that represents the waiting time between rare events. Using some mathematics, it can be shown that x has an exponential distribution. Let x > 0 be a random variable and let β > 0 be a constant. Then y = 1/ βe-x/β is a curve representing the exponential distribution. Areas under this curve give us exponential probabilities.
If a and b are any numbers such that 0 6 a 6 b, then using some extra mathematics, it can be shown that the area under the curve above the interval [a, b] is
P(a < x < b) = e - a/β - e -b/β
Notice that by de finition, x cannot be negative, so, P(x < 0) = 0. The random variable x is called an exponential random variable. Using some more mathematics, it can be shown that the mean and standard deviation of x are
( = β and ( = β
The number e = 2.71828 . . . is used throughout probability, statistics and mathematics. The key ex is conveniently located on most calculators.
The Poisson and exponential distributions have a special relationship. Specifi cally, it can be shown that the waiting time between successive Poisson arrivals (i.e., successes or rare events) has an exponential distribution with β = 1/(, where l is the average number of Poisson successes (rare events) per unit of time. For more on this topic, please see Problem 20.
Fatal accidents on scheduled domestic passenger flights are rare events. I
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