Sums of Gamma Random Variables
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
(1) f(t) = f(t; m,() =
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,() =
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).
Proposition 1. If m > 0 and ( > 0 then
(3) = 1
which confirms that f(t) defined by (1) is a valid density function.
Proof. = = 1 since ((m) = . (
Proposition 2. If X has a gamma distribution with parameters m and (, then the mean of X is
(4) (X = E(X) = = m(
Proof. = = m( = m( = m( 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) E(X2) = = m(m+1)(2
The variance of X is
(6) ((X)2 = E((X - (X)2) = m(2
The standard deviation of X is
(7) (X = (
Proof. = = m(m+1)(2 = m(m+1)(2 = m(m+1)(2 where we have used ((m+2) = m(m+1)((m) and (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
(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
(1/c)f(t/c) =
which is equal to f(t; m,c(). (
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
(12) = B(m, q)
where
(12) B(m, q) = = 2
is the beta function. To see this first note that ((m)((q) = . Make 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 < u < 1}. So ((m)((q) = = ((m+q) and (12) follows. Next we show that
(13) * =
To see this note that tm-1 * tq-1 = . Make the change of variables s = tu. We get tm-1 * tq-1 = = and (13) follows. It follows from (11) and (13) that
f(t; m,() * f(t; q,() = * = = f(t; m+q,()
and the propostion follows. (
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) = =
Proof. One has
L(s) = [((m)]-1 = [((m)]-1
If one makes the change of variables u = (s + 1/()t one obtains
L(s) = [((m)]-1
= (1/(1 + (s))m[((m)]-1 =
This proves (14). (15) follows from the (14) and the fact that M(r) = L(-r). (
Let
(16) G(t) = G(t;m,() = =
be the cummulative distribution function of the gamma random variable X ~ ((m,() and let
(17) H(t) = H(t;m,() = 1 - G(t) =
be the complementary distribution function (or survival function). Let
(18) (m(t) =
be the upper incomplete gamma function and
(19) (m(t) =
be the lower incomplete gamma function.
Proposition 7.
(20) G(t;m,() =
(21) H(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). (
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