Fundamentals of Probability and Statistics for Engineers

which is widely tabulated, and

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. F urthermore, 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 F igures 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 0 for
and at for
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

In the above, ( ,u) is the in co mplete gamma function,

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

A number of important distributions are special cases of the gamma distribu-
tion. Two of these are discussed below in more detail.

Some Important Continuous Distributions 213

…†ˆ… 1 †!; … 7 : 54 †













ˆ

1, xˆ"1)/ >1.



FX…x†ˆ

Z x

0

fX…u†duˆ



…†

Zx

0

u^1 eudu;

ˆ

…;x†

…†

; forx 0 ;

ˆ 0 ; elsewhere:

… 7 : 55 †



…;u†ˆ

Zu

0

x^1 exdx; … 7 : 56 †

mXˆ





;^2 Xˆ



^2

… 7 : 57 †