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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
ββ
xfor
x
xf ,
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
0,0,10,)1(
)()(
)(
)(
11
>><<−⋅⋅
Γ⋅Γ
+Γ
=
−−
βα
βα
β
βα
xforxxxf
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
>>>⋅−⋅⋅⋅=
−
βααβα
ββ
xforxxxf
While the corresponding cdf is given by
0,0,0),exp(1)( >>>⋅−−= βαα
β
xforxxF
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
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