The Poisson DistributionSpecification of the Poisson DistributionThe Normal Approximation to the PoissonInference for a Poisson distributionApproximate Confidence Interval for The `Exact' (Conservative) Confidence Interval for The Poisson ProcessIndependent Poisson Random VariablesA Comment on the Assumption of a Poisson ProcessSummaryPractice ProblemsSolutions to Practice ProblemsHomework ProblemsChapter 13The Poisson DistributionJeanne Antoinette Poisson (1721–1764), Marquise de Pompadour, was a member of the Frenchcourt and was the official chief mistress of Louis XV from 1745 until her death. The pompadourhairstyle was named for her. In addition, poisson is French for fish. The Poisson distribution,however, is named for Simeon-Denis Poisson (1781–1840), a French mathematician, geometerand physicist.13.1 Specification of the Pois son DistributionIn this chapter we will study a family of probability distributions for a countably infinite samplespace, each member of which is called a Poisson distributi on. Recall th at a binomial distributionis characterized by the values of two parameters: n and p. A Poisson distribution is simpler in thatit has only one parameter, which we denote by θ, pronounced theta. (Many books and websit esuse λ, pronounced lambda, instead of θ. We save λ for a related purpose.) The parameter θ mustbe positive: θ > 0. Below is the formul a for computing probabilities for the Poisson.P (X = x) =e−θθxx!, for x = 0, 1, 2, 3, . . . . (13.1)In this equation, e is th e famous number from calculus,e = limn→∞(1 + 1/n)n= 2.71828 . . . .You might recall, from the study of infinite series in calculus, that∞Xx=0bx/x! = eb,for any real number b. Thus,∞Xx=0P (X = x) = e−θ∞Xx=0θx/x! = e−θeθ= 1.321Table 13.1: A comparison of three probability distributions.Distribution o f X is:Poisson(1) Bin(1000, 0.001) Bin(500, 0.002)Mean : 1 1 1Variance : 1 0.999 0.998x P (X = x) P (X = x) P (X = x)0 0.3679 0.3677 0.367510.3679 0.3681 0.36822 0.1839 0.1840 0.18413 0.0613 0.0613 0.061340.0153 0.0153 0.01535 0.0031 0.0030 0.003060.0005 0.0005 0.0005≥ 7 0.0001 0.0001 0.0001Total 1.0000 1.0000 1.0000Thus, we s ee that Formula 13.1 is a mathemat ically valid way to assign probabili ties to the non-negative integers; i.e., all probabilit ies are nonnegative—indeed, they are positive—and they s umto one.The m ean of the Poiss on is its parameter θ; i.e., µ = θ. This can be proven using calculus and asimilar argument shows that the variance of a Poisson i s also equal to θ; i.e., σ2= θ and σ =√θ.When I write X ∼ Poi sson(θ) I mean that X is a random variable with its probability distribu-tion given by the Poisson d istribution with parameter value θ.I ask you for pati ence. I am going to delay my explanation of why the Poisson distribution i simportant in science.As we will see, the Poisson distribution is closely tied to the binomial. For example, let’s spenda few minutes looking at the three probability distributions presented in Table 13.1.There is a wealt h of useful informati on in this tabl e. In particular,1. If you were distressed that a Poisson random variable has an in fini te n u mber of possiblevalues—namely, every nonnegative integer—agonize n o longer! We see from the table thatfor θ = 1, 99.99% of the Poiss on probability is assigned to the event (X ≤ 6).2. If you read down the t hree colum ns of probabilities, you will see that the entries are nearlyidentical. Certainly, any one column of probabilities provides good approximations t o theentries in any other column . Thus, in some situations, a Poisson distribution can be used asan approximation to a binomial distribution.3. What do we need for the Poisson to be a good approximation to a binomial? First, we needto have the means of the distributions match; i.e., we need to use the Poisson with θ = np,as I did in Table 13. 1. The variance of a binomi al npq is necessarily smal ler than the mean322np because q < 1. Thu s, th e variance of a binomial cannot be made to match the variance ofthe Poisson:Variance of binomial = npq < np = θ = variance of Poisson.If, however, p is very close to 0, then q is very close to one and the variances almost matchas illustrated in Table 13.1.I will summarize the above observations in the following result.Result 13.1 (The Poisson approximation to the binomial.) The Bin(n, p) distri bution can be well-approximated by the Poisson(θ) di stribution if the foll owing cond itions are met:1. The distributions have the same mean; i.e., θ = np;2. The value of n is large and p is clo se to zero. In particular, the variance of the bi nomial npqshould be very close to the variance of the Poisson, θ = np.As a practical matter, we mostl y use this result if n > 1,000 because we can easily obtain exactbinomial probabilities from a website for n ≤ 1,000. Also, if np ≥ 25, our general guideline fromChapter 11 s tates that we may u se a Normal curve to obtain a good approximation t o th e binomial.Thus, again as a practical matter, we mostly use this result if θ = np ≤ 25, allowing us someindecision as to which approximation to use at np = 25, Normal or Poisson .Poisson probabilities can be computed by hand with a scientific calculator. Alternatively, thefollowing website can be used:http://stattrek.com/Tables/Poisson.aspx.I will give an example to i llustrate 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 ∼ Poiss on(10) andI am interested in P (X = 8). I go to the site and enter 8 in the box Poisson random variable, and Ienter 10 in the box Average rate of success. I click on the Calculate box and the site gives me thefollowing 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.As wi th our binomial calculator, there is a great deal of redundancy in these five answers.32313.1.1 The Normal Approximation to the PoissonPlease look at the Poisson(1) probabil ities in Table 13.1. We see that P (X = 0) = P (X = 1)and as x increases beyond 1, P (X = x) decreases. Thus, without actually drawing the probabil ityhistogram of the Poisson(1) we know th at i t is strongly skewed to the right; indeed, it has no lefttail! For θ < 1 the probability histogram is even more skewed than i t is for our
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