Applied Statistics and Probability for Engineers

4-10 ERLANG AND GAMMA DISTRIBUTIONS 131

It can be shown that the integral in the definition of is finite. Furthermore, by using inte-

gration by parts it can be shown that

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

Also, and it can be shown that. The gamma function can be in-

terpreted as a generalization to noninteger values of rof the term that is used in the

Erlang probability density function.

Now the gamma probability density function can be stated.

1 r 12!

 112  0 ! 1  11 22 
^1 2

 1 r 2  1 r 12!

 1 r 2  1 r 12  1 r 12

 1 r 2

Sketches of the gamma distribution for several values of and rare 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.



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 distributionis a special case of

The gamma functionis

 1 r 2  (4-19)



0

xr^1 ex dx, for r 0

Definition

The random variable Xwith probability density function

(4-20)

has a gamma random variablewith parameters. If ris an integer,

Xhas an Erlang distribution.

0 and r 0

f 1 x 2 

rxr^1 e x

 1 r 2

, for x 0

Definition

If Xis a gamma random variablewith parameters and r,

E 1 X 2 r  and 2 V 1 X 2 r ^2 (4-21)



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

function.

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