Page 52 - Computational Statistics Handbook with MATLAB

38 Computational Statistics Handbook with MATLAB

Gamma Distribution
1
λ = t = 1
0.9
0.8
λ = t = 3
0.7
0.6
f(x) 0.5

0.4
0.3 λ = t = 2
0.2
0.1
0
0 0.5 1 1.5 2 2.5 3
x

GU
8
8
F F F FI II U URE G 2. RE RE RE 2. 2. 2. 8 8
IG
GU
We show three examples of the gamma probability density function. We see that when
λ = t = 1 , we have the same probability density function as the exponential with parameter
λ = 1 .
The mean and variance of a chi-square random variable can be obtained from
the gamma distribution. These are given by
EX[] = , ν

and

VX() = 2ν .

W
eibul
WWeibuleibul ll l
Weibull
The Weibull distribution has many applications in engineering. In particular,
it is used in reliability analysis. It can be used to model the distribution of the
amount of time it takes for objects to fail. For the special case where ν = 0
⁄
and β = 1 , the Weibull reduces to the exponential with λ = 1 α .
The Weibull density for α > 0 and β > 0 is given by

© 2002 by Chapman & Hall/CRC