Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

3.3. TheΓ,χ^2 ,andβDistributions 177

One of the most important properties of the gamma distribution is its additive
property.

Theorem 3.3.1.LetX 1 ,...,Xnbe independent random variables. Suppose, for
i=1,...,n, thatXihas aΓ(αi,β)distribution. LetY=

∑n

i=1Xi.ThenYhas a

Γ(

∑n

i=1αi,β)distribution.

Proof:Using the assumed independence and the mgf of a gamma distribution, we

have by Theorem 2.6.1 that fort< 1 /β,

MY(t)=

∏n

i=1

(1−βt)−αi=(1−βt)−

Pn

i=1αi,

which is the mgf of a Γ(

∑n

i=1αi,β) distribution.

Γ-distributions naturally occur in the Poisson process, also.

Remark 3.3.1(Poisson Processes).Fort>0, letXtdenote the number of events
of interest that occur in the interval (0,t]. AssumeXtsatisfies the three assumptions
of a Poisson process. Letkbe a fixed positive integer and define the continuous
random variableWkto be the waiting time until thekthevent occurs. Then the
range ofWkis (0,∞). Note that forw>0,Wk>wif and only ifXw≤k−1.
Hence,

P(Wk>w)=P(Xw≤k−1) =

k∑− 1

x=0

P(Xw=x)=

k∑− 1

x=0

(λw)xe−λw

x!

.

In Exercise 3.3.5, the reader is asked to prove that

∫∞

λw

zk−^1 e−z

(k−1)!

dz=

k∑− 1

x=0

(λw)xe−λw

x!

.

Accepting this result, we have, forw>0, that the cdf ofWksatisfies

FWk(w)=1−

∫∞

λw

zk−^1 e−z

Γ(k)

dz=

∫λw

0

zk−^1 e−z

Γ(k)

dz,

and forw≤0,FWk(w) = 0. If we change the variable of integration in the integral
that definesFWk(w)bywritingz=λy,then

FWk(w)=

∫w

0

λkyk−^1 e−λy

Γ(k)

dy, w > 0 ,

andFWk(w)=0 forw≤0. Accordingly, the pdf ofWkis

fWk(w)=FW′ (w)=

{

λkwk−^1 e−λw

Γ(k)^0 <w<∞

0elsewhere.