Page 50 - Computational Statistics Handbook with MATLAB

36 Computational Statistics Handbook with MATLAB

In the Computational Statistics Toolbox, we include a function called csex-
λ
poc(x, ) that calculates the exponential cumulative distribution function
using Equation 2.34.

Gammm mmaa
GGaamm ma a
Ga
The gamma probability density function with parameters λ > 0 and t > 0 is
(
,
-------------------------------;
f x λ t) = λe – λx ( λx) t – 1 x ≥ 0, (2.36)
;
Γ t()
where t is a shape parameter, and λ is the scale parameter. The gamma func-
tion Γ t() is defined as
∞
∫ e y t – 1 d . y (2.37)
y
–
Γ t() =
0
For integer values of t, Equation 2.37 becomes

Γ t() = ( t – 1)! . (2.38)

Note that for t = 1, the gamma density is the same as the exponential. When
t is a positive integer, the gamma distribution can be used to model the
amount of time one has to wait until t events have occurred, if the inter-
arrival times are exponentially distributed.
The mean and variance of a gamma random variable are

t
EX[] = --- ,
λ

and

t
VX() = ----- .
λ 2

The cumulative distribution function for a gamma random variable is calcu-
lated using [Meeker and Escobar, 1998; Banks, et al., 2001]

© 2002 by Chapman & Hall/CRC