Introduction to Probability and Statistics for Engineers and Scientists

184 Chapter 5: Special Random Variables

=

(

λ

λ−t

)α 1 (

λ

λ−t

)α 2

from Equation 5.7.2

=

(

λ

λ−t

)α 1 +α 2

which is seen to be the moment generating function of a gamma (α 1 +α 2 ,λ) random
variable. Since a moment generating function uniquely characterizes a distribution, the
result entails.
The foregoing result easily generalizes to yield the following proposition.

PROPOSITION 5.7.1 IfXi,i =1,...,nare independent gamma random variables with
respective parameters (αi,λ), then

∑n

i= 1 Xiis gamma with parameters

∑n

i= 1 αi,λ.

0.07

0.06

0.05

0.04

0.03

0.02

0.01

(^003040506070)
(b)
012 3456789101112
(a)
0.6
0.5
0.4
0.3
0.2
0.1
a = 3
a = 4
a = 5
a = 0.5
a = 2
FIGURE 5.11 Graphs of the gamma (α,1) density for (a)α=.5, 2, 3, 4, 5and (b)α= 50.