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222 Chapter 4 Special Distributions by the same pdf). That said, it makes sense to single out these âreal-worldâ pdfs and investigate their properties in more detail. This, of course, is not an idea we are seeing for the ï¬rst timeârecall the attention given to the binomial and hypergeometric distributions in Section 3.2. Chapter 4 continues in the spirit of Section 3.2 by examining ï¬ve other widely used models. Three of the ï¬ve are discrete; the other two are continuous. One of the continuous pdfs is the normal (or Gaussian) distribution, which, by far, is the most important of all probability models. As we will see, the normal âcurveâ ï¬gures prominently in every chapter from this point on. Examples play a major role in Chapter 4. The only way to appreciate fully the generality of a probability model is to look at some of its speciï¬c applications. Thus, included in this chapter are case studies ranging from the discovery of alpha-particle radiation to an early ESP experiment to an analysis of volcanic eruptions to counting bug parts in peanut butter. 4.2 The Poisson Distribution The binomial distribution problems that appeared in Section 3.2 all had relatively small values for n, so evaluating p X (k) = P(X = k) = nk p k (1 â p)nâk was not particularly difï¬cult. But suppose n were 1000 and k, 500. Evaluating p X (500) would be a formidable task for many handheld calculators, even today. Two hundred years ago, the prospect of doing cumbersome binomial calculations by hand was a catalyst for mathematicians to develop some easy-to-use approximations. One of the ï¬rst such approximations was the Poisson limit, which eventually gave rise to the Poisson distribution. Both are described in Section 4.2. Simeon Denis Poisson (1781â1840) was an eminent French mathematician and physicist, an academic administrator of some note, and, according to an 1826 letter from the mathematician Abel to a friend, a man who knew âhow to behave with a great deal of dignity.â One of Poissonâs many interests was the application of probability to the law, and in 1837 he wrote Recherches sur la Probabilite de Jugements. Included in the latter is a limit for p X (k) = nk p k (1 â p)nâk that holds when n approaches â, p approaches 0, and np remains constant. In practice, Poissonâs limit is used to approximate hard-to-calculate binomial probabilities where the values of n and p reï¬ect the conditions of the limitâthat is, when n is large and p is small. The Poisson Limit Deriving an asymptotic expression for the binomial probability model is a straightforward exercise in calculus, given that np is to remain ï¬xed as n increases. Theorem 4.2.1 Suppose X is a binomial random variable, where n P(X = k) = p X (k) = p k (1 â p)nâk , k = 0, 1, . . . , n k If n â â and p â 0 in such a way that Î» = np remains constant, then n eânp (np)k p k (1 â p)nâk = lim P(X = k) = lim nââ nââ k! k pâ0 pâ0 np = const. np = const.