Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

3.7.∗Mixture Distributions 221

for−∞<x<∞, 0 <θ<∞. Therefore, the marginal (unconditional) pdfh(x)

ofXis found by integrating outθ;thatis,

h(x)=

∫∞

0

θα+1/^2 −^1

βα

√

2 πΓ(α)

exp

[

−θ

(

x^2

2

+

1

β

)]

dθ.

By comparing this integrand with a gamma pdf with parametersα+^12 and [(1/β)+
(x^2 /2)]−^1 , we see that the integral equals

h(x)=

Γ(α+^12 )

βα

√

2 πΓ(α)

(

2 β

2+βx^2

)α+1/ 2

, −∞<x<∞.

It is interesting to note that ifα=r/2andβ=2/r,whereris a positive integer,

thenXhas an unconditional distribution, which is Student’st,withrdegrees of

freedom. That is, we have developed a generalization of Student’s distribution

through this type of mixing or compounding. We note that the resulting distribution

(a generalization of Student’st) has much thicker tails than those of the conditional

normal with which we started.

The next two examples offer two additional illustrations of this type of com-

pounding.

Example 3.7.3.Suppose that we have a binomial distribution, but we are not

certain about the probabilitypof success on a given trial. Supposephas been

selected first by some random process that has a beta pdf with parametersαand

β.ThusX, the number of successes onnindependent trials, has a conditional

binomial distribution so that the joint pdf ofXandpis

p(x|p)g(p)=

n!

x!(n−x)!

px(1−p)n−x

Γ(α+β)

Γ(α)Γ(β)

pα−^1 (1−p)β−^1 ,

forx=0, 1 ,...,n, 0 <p< 1 .Therefore, the unconditional pmf ofXis given by

the integral

h(x)=

∫ 1

0

n!Γ(α+β)

x!(n−x)!Γ(α)Γ(β)

px+α−^1 (1−p)n−x+β−^1 dp

=

n!Γ(α+β)Γ(x+α)Γ(n−x+β)

x!(n−x)!Γ(α)Γ(β)Γ(n+α+β)

,x=0, 1 , 2 ,...,n.

Now supposeαandβare positive integers; since Γ(k)=(k−1)!, this unconditional
(marginal or compound) pdf can be written

h(x)=

n!(α+β−1)!(x+α−1)!(n−x+β−1)!

x!(n−x)!(α−1)!(β−1)!(n+α+β−1)!

,x=0, 1 , 2 ,...,n.

Because the conditional meanE(X|p)=np, the unconditional mean isnα/(α+β)

sinceE(p) equals the meanα/(α+β) of the beta distribution.