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4-10 ERLANG AND GAMMA DISTRIBUTIONS 131

Therefore, to define a gamma random variable, we require a generalization of the factorial
function.

Definition
The gamma function is

r 1 x
1r2 x e dx, for r 0 (4-19)

0

It can be shown that the integral in the definition of 1r2 is finite. Furthermore, by using inte-
gration by parts it can be shown that

1r2 1r 12 1r 12

This result is left as an exercise. Therefore, if r is a positive integer (as in the Erlang distribution),

1r2 1r 12!

Also, 112 0! 1 and it can be shown that 11 22
1 2 . The gamma function can be in-
terpreted as a generalization to noninteger values of r of the term 1r 12! that is used in the
Erlang probability density function.
Now the gamma probability density function can be stated.

Definition
The random variable X with probability density function

r r 1 x
x e
f 1x2 , for x 0 (4-20)
1r2
has a gamma random variable with parameters 0 and r 0 . If r is an integer,
X has an Erlang distribution.

Sketches of the gamma distribution for several values of and r are shown in Fig. 4-26. It can
be shown that f(x) satisfies the properties of a probability density function, and the following
result can be obtained. Repeated integration by parts can be used, but the details are lengthy.

If X is a gamma random variable with parameters and r,

2
E1X2 r and V1X2 r 2 (4-21)

Although the gamma distribution is not frequently used as a model for a physical system,
the special case of the Erlang distribution is very useful for modeling random experiments. The
exercises provide illustrations. Furthermore, the chi-squared distribution is a special case of