Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

160 Some Special Distributions

Ifr=1,thenYhas the pmf

pY(y)=p(1−p)y,y=0, 1 , 2 ,..., (3.1.4)

zero elsewhere, and the mgfM(t)=p[1−(1−p)et]−^1. In this special case,r=1,
we say thatY has ageometric distribution. In terms of Bernoulli trials,Y is
the number of failures until the first success. The geometric distribution was first
discussed in Example 1.6.3 of Chapter 1. For the last example, the probability
that exactly 11 patients have to have their blood type determined before the first
patient with type B blood is found is given by. 88110 .12. This is computed in R by
dgeom(11,0.12) = 0.0294.

3.1.2 MultinomialDistribution

The binomial distribution is generalized to the multinomial distribution as follows.
Let a random experiment be repeatednindependent times. On each repetition,
there is one and only one outcome from one ofkcategories. Call the categories
C 1 ,C 2 ,...,Ck. For example, the upface of a roll of a six-sided die. Then the
categories areCi={i},i=1, 2 ,...,6. Fori=1,...,k,letpibe the probability that
the outcome is an element ofCiand assume thatpiremains constant throughout
thenindependent repetitions. Define the random variableXito be equal to the
number of outcomes that are elements ofCi,i=1, 2 ,...,k−1. BecauseXk=
n−X 1 −···−Xk− 1 ,Xkis determined by the otherXi’s. Hence, for the joint
distribution of interest we need only considerX 1 ,X 2 ,...,Xk− 1.
The joint pmf of (X 1 ,X 2 ,...,Xk− 1 )is

P(X 1 =x,X 2 =x 2 ,...,Xk− 1 =xk− 1 )=

n!

x 1 !···xk− 1 !xk!

px 11 ···pxkk−− 11 pxkk, (3.1.5)

for allx 1 ,x 2 ,...,xk− 1 that are nonnegative integers and such thatx 1 +x 2 +···+
xk− 1 ≤n,wherexk=n−x 1 −···−xk− 1 andpk=1−

∑k− 1
j=1pj.Wenextshow
that expression (3.1.5) is correct. The number of distinguishable arrangements of
x 1 C 1 s,x 2 C 2 s,...,xkCksis
(
n
x 1

)(

n−x 1

x 2

)

···

(

n−x 1 −···−xk− 2

xk− 1

)

=

n!

x 1 !x 2 !···xk!

and the probability of each of these distinguishable arrangements is

px 11 px 22 ···pxkk.

Hence the product of these two latter expressions gives the correct probability, which
is in agreement with expression (3.1.5).
We say that (X 1 ,X 2 ,...,Xk− 1 )hasamultinomial distributionwith param-
etersnandp 1 ,...,pk− 1. The joint mgf of (X 1 ,X 2 ,...,Xk− 1 )isM(t 1 ,...,tk− 1 )=

E(exp{

∑k− 1

i=1tiXi}), i.e.,

M(t 1 ,...,tk− 1 )=

∑

···

∑ n!

x 1 !···xk− 1 !xk!

(p 1 et^1 )x^1 ···(pk− 1 etk−^1 )xk−^1 pxkk,