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The corresponding (cumulative) distribution function (cdf) would be given by
an integral that has no closed-form solution.
The exponential distribution
The exponential distribution is the gamma distribution with a = 1. Its pdf is
given by
0
,
0
),
exp(
1
)
(
>
>
−
⋅
=
β
β
β
x
for
x
x
f
,
while its cdf is given by F(x) = 1 - exp(-x/
β
), for x>0,
β
>0.
The beta distribution
The pdf for the gamma distribution is given by
0
,
0
,
1
0
,
)
1
(
)
(
)
(
)
(
)
(
1
1
>
>
<
<
−
⋅
⋅
Γ
⋅
Γ
+
Γ
=
−
−
β
α
β
α
β
α
β
α
x
for
x
x
x
f
As in the case of the gamma distribution, the corresponding cdf for the beta
distribution is also given by an integral with no closed-form solution.
The Weibull distribution
The pdf for the Weibull distribution is given by
0
,
0
,
0
),
exp(
)
(
1
>
>
>
⋅
−
⋅
⋅
⋅
=
−
β
α
α
β
α
β
β
x
for
x
x
x
f
While the corresponding cdf is given by
0
,
0
,
0
),
exp(
1
)
(
>
>
>
⋅
−
−
=
β
α
α
β
x
for
x
x
F
Functions for continuous distributions
To define a collection of functions corresponding to the gamma, exponential,
beta, and Weibull distributions, first create a sub-directory called CFUN