2.2
The Gamma Distribution
In this section we look at some of the basic properties of gamma random variables; see
Hogg and Tanis [6].
A random variable X is said to have a gamma distribution with parameters m > 0 and
 > 0 if its probability density function has the form
f(t) = f(t; m,) =
(1)
tm-1e-t/
m
 (m)


0
if t > 0
if t < 0
In this case we shall say X is a gamma random variable with parameters m and  and
write X ~ (m,). Sometimes m is called the shape parameter and  the scale parameter.
In general, m might not be an integer.
Gamma random variables are used to model a number of physical quantities. Some
examples are
1.
The time it takes for something to occur, e.g. a lifetime or the service time
in a queue.
2.
The rate at which some physical quantity is accumulating during a certain
period of time, e.g. the excess water flowing into a dam during a certain
period of time due to rain or the amount of grain harvested during a certain
season.
Sometimes it is convenient to use  = 1/ as a parameter instead of . The pdf then has
the form
(2)
f(t) = f(t; m,) =
mtm-1e-t
 (m)
for t > 0. We shall also write X ~ (m,1/) in this case. It should be clear from the
context whether f(t; m,) or f(t; m,) stands for (1) or (2).
2.2 - 1
Proposition 1. If m > 0 and  > 0 then

m-1 -t/
 t m e dt
   (m)
(3)
= 1
0
which confirms that f(t) defined by (1) is a valid density function.


0
0

m-1 -t/
um-1e-u
 t e 
m-1 -u

Proof.  m
dt = 
du
=
1
since

(m)
=
  (m)
 u e du. 

   (m)
0
Proposition 2. If X has a gamma distribution with parameters m and , then the mean of
X is

m-1 -t/
t e
X = E(X) =  t  m (m) dt = m

(4)
0

Proof.  t


m -t/
m-1 -t/
t e
 t m e  dt = 
 m (m) dt
  (m)

0
0


tme-t/

= m  m+1
dt = m 
 f(t;m+1,) dt = m
   (m+1)
0
0
where we have used  (m+1) = m (m) and (3). 
Proposition 3. If X has a gamma distribution with parameters m and , then the
expected value of X2 is

(5)
m-1 -t/
t e
E(X ) =  t2  m (m) dt

 
2
= m(m+1)2
0
The variance of X is
(6)
(X)2 = E((X - X)2) = m2
The standard deviation of X is
2.2 - 2
X =
(7)
m



tm+1e-t/
 t e
2 t e
2


Proof.
t
dt =  m
dt = m(m+1)  m+2
dt =
  m (m)
   (m)
   (m+2)
m-1 -t/
0
m+1 -t/
0
0

2
m(m+1) 
 f(t;m+2,) dt = m(m+1) where we have used  (m+2) = m(m+1) (m) and
2
0
(3). This proves (5). Since E((X - X)2) = E(X2) - X2 the formula (6) follows from (4)
and (5). (7) follows from (6). 
Proposition 4. If f(t) is given by (1) then for t > 0 one has
[(m - 1) - t] tm-2e-t/
m+1 (m)
(8)
f '(t) =
(9)
f(t) has a single local maximum at t = (m - 1) if m > 1.
(10)
f(t) is strictly decreasing for t > 0 if m  1
Proof. (8) is a straightforward computation and (9) and (10) follow from (8). 
Proposition 5. Assume X has a gamma distribution with parameters m and  and let
Y = cX for some positive number c. Then Y has a gamma distribution with parameters m
and c.
Proof. If f(t) given by (1) is the density function of X then the density function of Y is
tm-1e-t/(c)
 (c)  (m)

0
m
(1/c)f(t/c)
=
if t > 0
if t < 0
which is equal to f(t; m,c). 
2.2 - 3
Proposition 5. If X and Y are independent gamma random variables and X has
parameters m and  and Y has parameters q and , then X + Y is a gamma random
variable with parameters m + q and .
Proof. We first show that
1
(12)
 um-1(1-u)q-1 du = B(m, q)

0
where
(12)
B(m, q) =
/2
 (m) (q)
2m-1
2q-1
= 2
 cos (t) sin (t) dt
 (m +q )
0


0
0
m-1 q-1 -(t+s)
is the beta function. To see this first note that  (m) (q) = 
dsdt. Make
 
t s e
the change of variables r = t + s and u = t/(t+s). Then t = ru and s = r(1-u) and dsdt =
rdudr and the first quadrant in the st-plane gets mapped into the strip {(r,u): 0 < r < , 0

1
 
 (ru)
< u < 1}. So  (m) (q) = 
m-1
0
1
 um-1(1-u)q-1 du
(r(1-u)) e rdudr =  (m +q ) 
q-1 -r
0
0
and (12) follows. Next we show that
(13)
m-1
q-1
 t
  t


*

  (m)    (q) 
=
tm+q-1
 (m +q)
t
To see this note that t
m-1
*t
q-1
 sm-1(t-s)q-1 ds. Make the change of variables s = tu.
= 
0
1
1
m-1
q-1
m+q-1
m-1
q-1
m+q-1  (m) (q)

We get tm-1 * tq-1 = 
(tu)
(t-tu)
tdu
=
t

 u (1-u) du = t
 (m +q )
0
0
and (13) follows. It follows from (11) and (13) that
tm+q-1e-t/
 tm-1e-t/   tq-1e-t/ 
f(t; m,) * f(t; q,) =  m
= f(t; m+q,)
* q
 = m+q
  (m +q)
  (m)   (q)
and the propostion follows. 
2.2 - 4
Proposition 6. If X has a gamma distribution with parameters m and  = 1/, then the
Laplace transform L(s) and moment generating function M(r) of X are given by
(14)
L(s) =
(15)
M(r) =
1

=
(1 + s)m
( + s)m
1
(1 - r)
m
=

( - s)m
Proof. One has


0
0
-st -m m-1 -t/
-1
-m m-1 -(s+1/)t
L(s) = [ (m)]-1
dt
 e  t e dt = [ (m)] 
  t e
If one makes the change of variables u = (s + 1/)t one obtains

 -m(u/(s + 1/))m-1e-u (1/(s + 1/))du
L(s) = [ (m)] 
-1
0

1
 um-1e-u du =
= (1/(1 + s)) [ (m)] 
(1 + s)m
m
-1
0
This proves (14). (15) follows from the (14) and the fact that M(r) = L(-r). 
Let
t
t
(16)
m-1 -s/
s e
 f(s;m,) ds =  m
G(t) = G(t;m,) = 
ds

  (m)
0
0
be the cummulative distribution function of the gamma random variable X ~ (m,) and
let

(17)
m-1 -s/
s e
H(t) = H(t;m,) = 1 - G(t) =   m (m) ds

t
be the complementary distribution function (or survival function). Let

(18)
m-1 -s
m(t) = 
 s e ds
t
be the upper incomplete gamma function and
t
(19)
m-1 -s
m(t) = 
 s e ds
0
be the lower incomplete gamma function.
2.2 - 5
Proposition 7.
 (t/)
(20)
G(t;m,) = m (m)
(21)
H(t;m,) =
m(t/)
 (m)
Proof. (20) follows by making the change of variables u = s/ in (16) and (21) follows
by making the change of variables u = s/ in (17). 
2.2 - 6