6More distributions

IA Probability


6.2 Gamma distribution
Suppose
X
1
, ··· , X
n
are iid
E
(
λ
). Let
S
n
=
X
1
+
···
+
X
n
. Then the mgf of
S
n
is
E
h
e
θ(X
1
+...+X
n
)
i
= E
e
θX
1
···E
e
θX
n
=
E
e
θX

n
=
λ
λ − θ
n
.
We call this a gamma distribution.
We claim that this has a distribution given by the following formula:
Definition (Gamma distribution). The gamma distribution Γ(n, λ) has pdf
f(x) =
λ
n
x
n−1
e
−λx
(n − 1)!
.
We can show that this is a distribution by showing that it integrates to 1.
We now show that this is indeed the sum of n iid E(λ):
E[e
θX
] =
Z
∞
0
e
θx
λ
n
x
n−1
e
−λx
(n − 1)!
dx
=
λ
λ − θ
n
Z
∞
0
(λ − θ)
n
x
n−1
e
−(λ−θ)x
(n − 1)!
dx
The integral is just integrating over Γ(n, λ − θ), which gives one. So we have
E[e
θX
] =
λ
λ − θ
n
.