Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

3.2. The Poisson Distribution 167

3.1.29.Let the independent random variablesX 1 andX 2 have binomial distri-
butions with parametersn 1 ,p 1 =^12 andn 2 ,p 2 =^12 , respectively. Show that
Y=X 1 −X 2 +n 2 has a binomial distribution with parametersn=n 1 +n 2 ,p=^12.

3.1.30. Consider a shipment of 1000 items into a factory. Suppose the factory
can tolerate about 5% defective items. LetX be the number of defective items
in a sample without replacement of sizen= 10. Suppose the factory returns the
shipment ifX≥2.

(a)Obtain the probability that the factory returns a shipment of items that has

5% defective items.

(b)Suppose the shipment has 10% defective items. Obtain the probability that

the factory returns such a shipment.

(c)Obtain approximations to the probabilities in parts (a) and (b) using appro-

priate binomial distributions.

Note:If you do not have access to a computer package with a hypergeometric com-
mand, obtain the answer to (c) only. This is what would have been done in practice
20 years ago. If you have access to R, then the commanddhyper(x,D,N-D,n)
returns the probability in expression (3.1.7).

3.1.31.Show that the variance of a hypergeometric (N, D, n) distribution is given
by expression (3.1.8).
Hint:First obtainE[X(X−1)] by proceeding in the same way as the derivation of
the mean given in Section 3.1.3.

3.2 ThePoissonDistribution

Recall that the following series expansion^3 holds for all real numbersz:

1+z+

z^2

2!

+

z^3

3!

+···=

∑∞

x=0

zx

x!

=ez.

Consider the functionp(x) defined by

p(x)=

{ λxe−λ

x! x=0,^1 ,^2 ,...

0elsewhere,

(3.2.1)

whereλ>0. Sinceλ>0, thenp(x)≥0and

∑∞

x=0

p(x)=

∑∞

x=0

λxe−λ

x!

=e−λ

∑∞

x=0

λx

x!

=e−λeλ=1;

(^3) See, for example, the discussion on Taylor series inMathematical Commentsreferenced in the
Preface.