Page 311 - Applied Probability

14. Poisson Approximation
300
in I [1]. Here B α is a subset of I containing α. Usually B α is chosen so that
X α is independent of those X β with β outside B α . If this is the case, then
define two constants

=
b 1
α∈I β∈B α p α p β

b 2 = p αβ ,
α∈I β∈B α\{α}
where
p αβ =E(X α X β )
=Pr(X α =1,X β =1).
In this context, λ − Var(S)= b 1 − b 2 , and the total variation distance
between S and its Poisson approximation Z with mean λ satisfies
1 − e −λ
L(S) −L(Z) Ø (b 1 + b 2 ). (14.3)
λ
Both Chen-Stein methods are well adapted to solving many problems
arising in mathematical genetics. We will illustrate the main ideas through
a sequence of examples. Readers interested in mastering the underlying
theory are urged to consult the references [2, 5, 15].

14.2 The Law of Rare Events

Suppose that X 1 ,... ,X n are independent Bernoulli random variables with
success probabilities p 1 ,...,p n. If the p α are small and the mean number
n

of successes λ = p α is moderate in size, then the law of rare events
α=1 n
declares that the sum S = X α is approximately Poisson distributed.
α=1
The neighborhood method provides an easy verification of this result. If we
let N α be the singleton set {α} and Z be a Poisson random variable with
mean λ, then inequality (14.3) reduces to
n
1 − e −λ 2
π S − π Z TV ≤ p α
λ
α=1
because the sum p αβ is empty.
β∈N α\{α}

14.3 Poisson Approximation to the W Statistic
d

In Chapter 4 we studied the W d statistic for multinomial trials. Recall that
W d denotes the number of categories with d or more successes after n trials.