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371chapter4f2011 - Chapter 4 The Poisson Distribution 4.1...

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Chapter 4 The Poisson Distribution 4.1 The Fish Distribution? The Poisson distribution is named for Simeon-Denis Poisson (1781–1840). In addition, poisson is French for fish. In this chapter we will study a family of probability distributions for a countably infinite sample space, each member of which is called a Poisson Distribution . Recall that a binomial distribution is characterized by the values of two parameters: n and p . A Poisson distribution is simpler in that it has only one parameter, which we denote by θ , pronounced theta . (Many books and websites use λ , pronounced lambda, instead of θ . We save λ for a related purpose.) The parameter θ must be positive: θ > 0 . Below is the formula for computing probabilities for the Poisson. P ( X = x ) = e - θ θ x x ! , for x = 0 , 1 , 2 , 3 , . . .. (4.1) In this equation, e is the famous number from calculus, e = lim n →∞ (1 + 1 /n ) n = 2 . 71828 . . . . You might recall from the study of infinite series in calculus, that x =0 b x /x ! = e b , for any real number b . Thus, x =0 P ( X = x ) = e - θ x =0 θ x /x ! = e - θ e θ = 1 . Thus, we see that Formula 4.1 is a mathematically valid way to assign probabilities to the nonneg- ative integers. The mean of the Poisson is its parameter θ ; i.e. μ = θ . This can be proven using calculus and a similar argument shows that the variance of a Poisson is also equal to θ ; i.e. σ 2 = θ and σ = θ . 43
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When I write X Poisson( θ ) I mean that X is a random variable with its probability distribu- tion given by the Poisson with parameter value θ . I ask you for patience. I am going to delay my explanation of why the Poisson distribution is important in science. Poisson probabilities can be computed by hand with a scientific calculator. Alternatively, the following website, which is linked to our course webpage, can be used: http://stattrek.com/Tables/Poisson.aspx I will give an example to illustrate the use of this site. Let X Poisson( θ ). The website calculates five probabilities for you: P ( X = x ); P ( X < x ); P ( X x ); P ( X > x ); and P ( X x ) . You must give as input your value of θ and a value of x . Suppose that I have X Poisson(10) and I am interested in P ( X = 8) . I go to the site and type ‘8’ in the box labeled ‘Poisson random variable,’ and I type ‘10’ in the box labeled ‘Average rate of success.’ I click on the ‘Calculate’ box and the site gives me the following answers: P ( X = 8) = 0 . 1126; P ( X < 8) = 0 . 2202; P ( X 8) = 0 . 3328; P ( X > 8) = 0 . 6672; and P ( X 8) = 0 . 7798 . (There is, of course, a great deal of redundancy in these five answers because two pairs of events are complements of each other.) It can be shown that for the Poisson, if θ 5 then its probability histogram is markedly asymmetrical, but if θ 25 its probability histogram is approximately symmetric and bell-shaped. This last statement suggests that we might use a normal curve to compute approximate probabilities for the Poisson, provided θ is large.
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