Page 230 - Fundamentals of Probability and Statistics for Engineers

Some Important Continuous Distributions 213

which is widely tabulated, and
†ˆ 1†!; 7:54†

when is a positive integer.
The parameters associated with the gamma distribution are and ; both
are taken to be positive. Since the gamma distribution is one-sided, physical
quantities that can take values only in, say, the positive range are frequently
modeled by it. Furthermore, it serves as a useful model because of its versatility
in the sense that a wide variety of shapes to the gamma density function can be

obtained by varying the values of and . This is illustrated in Figures 7.10(a)
and 7.10(b) which show plots of Equation (7.52) for several values of and .
We notice from these figures that determines the shape of the distribution and
is thus a shape parameter whereas is a scale parameter for the distribution. In
general, the gamma density function is unimodal, with its peak at x ˆ 0 for
1, and at x ˆ " 1)/ for > 1.
As we will verify in Section 7.4.1.1, it can also be shown that the gamma
distribution is an appropriate model for time required for a total of exactly
Poisson arrivals. Because of the wide applicability of Poisson arrivals, the
gamma distribution also finds numerous applications.
The distribution function of random variable X having a gamma distribution is

Z x Z x
F X x†ˆ f u†du ˆ u 1 u du;
e
X
0 † 0
; x† 7:55†
ˆ ; for x 0;
†
ˆ 0; elsewhere:

In the above, ( , u) is the incomplete gamma function,

Z u
e
; u†ˆ x 1 x dx; 7:56†
0
which is also widely tabulated.
The mean and variance of a gamma-distributed random variable X take quite
simple forms. After carrying out the necessary integration, we obtain
2
m X ˆ ; ˆ 7:57†
X
2
A number of important distributions are special cases of the gamma distribu-
tion. Two of these are discussed below in more detail.

TLFeBOOK