Introduction to Probability and Statistics for Engineers and Scientists

*5.7The Gamma Distribution 183

we see that

(n)=(n−1)!

The function (α) is called thegammafunction.
It should be noted that whenα=1, the gamma distribution reduces to the exponential
with mean 1/λ.
The moment generating function of a gamma random variableXwith parameters (α,λ)
is obtained as follows:

φ(t)=E[etX]

=

λα

(α)

∫∞

0

etxe−λxxα−^1 dx

=

λα

(α)

∫∞

0

e−(λ−t)xxα−^1 dx

=

(

λ

λ−t

)α

1

(α)

∫∞

0

e−yyα−^1 dy [byy=(λ−t)x]

=

(

λ

λ−t

)α

(5.7.2)

Differentiation of Equation 5.7.2 yields

φ′(t)=

αλα

(λ−t)α+^1

φ′′(t)=

α(α+1)λα

(λ−t)α+^2

Hence,

E[X]=φ′(0)=

α

λ

(5.7.3)

Var(X)=E[X^2 ]−(E[X])^2

=φ′′(0)−

(α

λ

) 2

=

α(α+1)

λ^2

−

α^2

λ^2

=

α

λ^2

(5.7.4)

An important property of the gamma is that ifX 1 andX 2 are independent gamma
random variables having respective parameters (α 1 ,λ) and (α 2 ,λ), thenX 1 +X 2 is a
gamma random variable with parameters (α 1 +α 2 ,λ). This result easily follows since

φX 1 +X 2 (t)=E[et(X^1 +X^2 )] (5.7.5)

=φX 1 (t)φX 2 (t)