n

We can derive the equation for computing Poisson probabilities directly from the bi- nomial probability distribution by taking the mathematical limits as P 0 and 1. With these limits, the parameter 1 = nP is a constant that specifies the average number of occurrences (successes) for a particular time and/or space. We can see intuitively that the Poisson is a special case of the binomial obtained by extending these limits. However, the mathematical derivation is beyond the scope of this book. The interested reader is referred to page 244 of Hogg and Craig (1995). The Poisson probability distribution function is given in Equation 4.21. 4.5 Poisson Distribution 147 The Poisson Distribution Function, Mean, and Variance The random variable X is said to follow the Poisson distribution if it has the probability distribution P(x) = *** x! for x = 0,1,2,... (4.21) where P(x) = the probability of x successes over a given time or space, given = the expected number of successes per time or space unit, 1 > 0 2.71828 (the base for natural logarithms) The mean and variance of the Poisson distribution are My = E(X) = 1 and o; = E(X - M.)2) = x The sum of Poisson random variables is also a Poisson random variable. Thus, the sum of K Poisson random variables, each with mean X, is a Poisson random variable with mean KA. Two important applications of the Poisson distribution in the modern global economy are the probability of failures in complex systems and the probability of defective products in large production runs of several hundred thousand to a million units. A large world- wide shipping company such as Federal Express has a complex and extensive pickup, classification, shipping, and delivery system for millions of packages each day. There is a very small probability of handling failure at each step for each of the millions of packages handled every day. The company is interested in the probability of various numbers of failed deliveries each day when the system is operating properly. If the number of actual failed deliveries observed on a particular day has a small probability of occurring, given proper targeted operations, then the management begins a systematic checking process to identify and correct the reason for excessive failures. Example 4.10 System Component Failure (Poisson Probabilities) Andrew Whittaker, computer center manager, reports that his computer system expe- rienced three component failures during the past 100 days. a. What is the probability of no failures in a given day? b. What is the probability of one or more component failures in a given day? c. What is the probability of at least two failures in a 3-day period