Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

3.1. The Binomial and Related Distributions 161

where the multiple sum is taken over all nonnegative integers and such thatx 1 +
x 2 +···+xk− 1 ≤n.Letm=

∑k− 1

i=1pie

ti+pk− 1. Recall thatxk=n−∑k−^1
i=1xi.
Then sincem>0, we have

M(t 1 ,...,tk− 1 )=mn

∑

···

∑ n!

x 1 !···xk− 1 !xk!

×

(

p 1 et^1

m

)x 1

···

(

pk− 1 etk−^1

m

)xk− 1 (

pk

m

)xk

= mn×1=

(k− 1

∑

i=1

pieti+pk− 1

)n

, (3.1.6)

where we have used the property that sum of a pmf over its support is 1.

We can use the joint mgf to determine marginal distributions. The mgf ofXiis

M(0,..., 0 ,ti, 0 ,...,0) = (pieti+(1−pi))n;

hence,Xiis binomial with parametersnandpi.Themgfof(Xi,Xj),i<j,is

M(0,..., 0 ,ti, 0 ,..., 0 ,tj, 0 ,...,0) = (pieti+pjetj+(1−pi−pj))n;

so that (Xi,Xj) has a multinomial distribution with parametersn,pi,andpj.At
times, we say that (X 1 ,X 2 )hasatrinomial distribution.
Another distribution of interest is the conditional distribution ofXigivenXj.
For convenience, we selecti=2andj= 1. We know that (X 1 ,X 2 ) is multinomial
with parametersnandp 1 andp 2 and thatX 1 is binomial with parametersnand
p 1. Thus, the conditional pmf is,

pX 2 |X 1 (x 2 |x 1 )=

pX 1 ,X 2 (x 1 ,x 2 )

pX 1 (x 1 )

=

x 1 !(n−x 1 )!

n!px 11 [1−p 1 ]n−x^1

n!px 11 px 22 [1−(p 1 +p 2 )]n−(x^1 +x^2 )

x 1 !x 2 ![n−(x 1 +x 2 )]!

=

(

n−x 1

x 2

)

px 22

(1−p 1 )x^2

[(1−p 1 )−p 2 ]n−x^1 −x^2

(1−p 1 )n−x^1 −x^2

=

(

n−x 1

x 2

)(

p 2

1 −p 1

)x 2 (

1 −

p 2

1 −p 1

)n−x 1 −x 2

,

for 0≤x 2 ≤n−x 1 .Notethatp 2 < 1 −p 1. Thus, the conditional distribution of
X 2 givenX 1 =x 1 is binomial with parametersn−x 1 andp 2 /(1−p 1 ).
Based on the conditional distribution ofX 2 givenX 1 ,wehaveE(X 2 |X 1 )=
(n−X 1 )p 2 /(1−p 1 ). Letρ 12 be the correlation coefficient betweenX 1 andX 2.
Since the conditional mean is linear with slope−p 2 /(1−p 1 ),σ 2 =

√

np 2 (1−p 2 ),

andσ 1 =

√

np 1 (1−p 1 ), it follows from expression (2.5.4) that

ρ 12 =−

p 2

1 −p 1

σ 1

σ 2

=−

√

p 1 p 2

(1−p 1 )(1−p 2 )

.

Because the support ofX 1 andX 2 has the constraintx 1 +x 2 ≤n, the negative

correlation is not surprising.