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222 Chapter 4 Special Distributions
by the same pdf). That said, it makes sense to single out these “real-world” pdfs and
investigate their properties in more detail. This, of course, is not an idea we are seeing for the first time—recall 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 five other widely
used models. Three of the five 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 “curve” figures
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 specific 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 − p)n−k was not particularly difficult. 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
first such approximations was the Poisson limit, which eventually gave rise to the
Poisson distribution. Both are described in Section 4.2.
Simeon Denis Poisson (1781–1840) 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 “how to behave
with a great deal of dignity.” One of Poisson’s 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 − p)n−k that holds when
n approaches ∞, p approaches 0, and np remains constant. In practice, Poisson’s
limit is used to approximate hard-to-calculate binomial probabilities where the values of n and p reflect the conditions of the limit—that 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 fixed 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 − p)n−k , k = 0, 1, . . . , n
k
If n → ∞ and p → 0 in such a way that λ = np remains constant, then
n e−np (np)k
p k (1 − p)n−k =
lim P(X = k) = lim
n→∞
n→∞
k!
k
p→0
p→0
np = const.
np = const.