Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

220 Some Special Distributions

The mixture of distributions is sometimes calledcompounding.Moreover,it
does not need to be restricted to a finite number of distributions. As demonstrated
in the following example, a continuous weighting function, which is of course a pdf,
can replacep 1 ,p 2 ,...,pk; i.e., integration replaces summation.

Example 3.7.2.LetXθbe a Poisson random variable with parameterθ.Wewant

to mix an infinite number of Poisson distributions, each with a different value ofθ.

We let the weighting function be a pdf ofθ, namely, a gamma with parametersα

andβ.Forx=0, 1 , 2 ,..., the pmf of the compound distribution is

p(x)=

∫∞

0

[

1

βαΓ(α)

θα−^1 e−θ/β

][

θxe−θ

x!

]

dθ

=

1

Γ(α)βαx!

∫∞

0

θα+x−^1 e−θ(1+β)/βdθ

=

Γ(α+x)βx

Γ(α)x!(1 +β)α+x

,

where the third line follows from the change of variablet=θ(1 +β)/βto solve the
integral of the second line.
An interesting case of this compound occurs whenα=r, a positive integer, and
β=(1−p)/p,where0<p<1. In this case the pmf becomes

p(x)=

(r+x−1)!

(r−1)!

pr(1−p)x

x!

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

That is, this compound distribution is the same as that of the number of excess
trials needed to obtainrsuccesses in a sequence of independent trials, each with
probabilitypof success; this is one form of thenegative binomial distribution.
The negative binomial distribution has been used successfully as a model for the
number of accidents (see Weber, 1971).

In compounding, we can think of the original distribution ofXas being a con-
ditional distribution givenθ, whose pdf is denoted byf(x|θ). Then the weighting
function is treated as a pdf forθ,sayg(θ). Accordingly, the joint pdf isf(x|θ)g(θ),
and the compound pdf can be thought of as the marginal (unconditional) pdf ofX,

h(x)=

∫

θ

g(θ)f(x|θ)dθ,

where a summation replaces integration in caseθhas a discrete distribution. For
illustration, suppose we know that the mean of the normal distribution is zero but
the varianceσ^2 equals 1/θ >0, whereθhas been selected from some random model.
For convenience, say this latter is a gamma distribution with parametersαandβ.
Thus, given thatθ,Xis conditionallyN(0, 1 /θ) so that the joint distribution ofX
andθis

f(x|θ)g(θ)=

[√

θ

√

2 π

exp

(

−θx^2

2

)][

1

βαΓ(α)

θα−^1 exp(−θ/β)

]

,