Mathematical Methods for Physics and Engineering : A Comprehensive Guide

PROBABILITY

0

0 246810 12 14 16 18 20

0. 2

0. 4

0. 6

0. 8

1

r=1

r=2

r=5

r=10

x

f(x)

Figure 30.16 The PDFf(x) for the gamma distributionsγ(λ, r)withλ=1

andr=1, 2 , 5 ,10.

defined for all positiverby replacing (r−1)! by Γ(r) in (30.118); see the appendix

for a discussion of the gamma function Γ(x). If a random variableXis described

by a gamma distribution of orderrwith parameterλ,wewriteX∼γ(λ, r);

we note that the exponential distribution is the special caseγ(λ,1). The gamma

distributionγ(λ, r) is plotted in figure 30.16 forλ= 1 andr=1, 2 , 5 ,10. For

larger, the gamma distribution tends to the Gaussian distribution whose mean

and variance are specified by (30.120) below.

The MGF for the gamma distribution is obtained from that for the exponential

distribution, by noting that we may consider the interval between everyrth event

in a Poisson process as the sum ofrintervals between successive events. Thus the

rth-order gamma variate is the sum ofrindependent exponentially distributed

random variables. From (30.117) and (30.90), the MGF of the gamma distribution

is therefore given by

M(t)=

(

λ

λ−t

)r

, (30.119)

from which the mean and variance are found to be

E[X]=

r

λ

,V[X]=

r

λ^2

. (30.120)

We may also use the above MGF to prove another useful theorem regarding

multiple gamma distributions. IfXi∼γ(λ, ri),i=1, 2 ,...,n, are independent

gamma variates then the random variableY=X 1 +X 2 +···+Xnhas MGF

M(t)=

∏n

i=1

(

λ

λ−t

)ri

=

(

λ

λ−t

)r 1 +r 2 +···+rn

. (30.121)

ThusYis also a gamma variate, distributed asY∼γ(λ, r 1 +r 2 +···+rn).