KUMCDF

KUMCDF (LET)

Library Function

Compute the Kumaraswamy cumulative distribution function with shape parameters and .

The standard Kumaraswamy distribution has the following cumulative distribution function:

with and denoting the shape parameters.

This distribution can be extended with lower and upper bound parameters. If a and b denote the lower and upper bounds, respectively, then the location and scale parameters are:

location = a
scale = b - a


The general form of the distribution can then be found by using the relation

This distribution has been proposed as a more tractable alternative to the beta distribution.

LET <y> = KUMCDF(<x>,<alpha>,<beta>,<a>,<b>)
                        <SUBSET/EXCEPT/FOR qualification>
where <x> is a number, parameter, or variable containing values in the interval (a,b);
            <y> is a variable or a parameter (depending on what <x> is) where the computed Kumaraswamy cdf value is stored;
            <alpha> is a positive number, parameter, or variable that specifies the first shape parameter;
            <beta> is a positive number, parameter, or variable that specifies the second shape parameter;
            <a> is a number, parameter, or variable that specifies the lower limit;
            <b> is a number, parameter, or variable that specifies the upper limit;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

If <a> and <b> are omitted, they default to 0 and 1, respectively.

LET A = KUMCDF(0.3,0.2,1.2)
LET Y = KUMCDF(X,0.5,2)
PLOT KUMCDF(X,2,3) FOR X = 0 0.01 1

None

None

KUMPDF	= Compute the Kumaraswamy probability density function.
KUMPPF	= Compute the Kumaraswamy percent point function.
RGTPDF	= Compute the reflected generalized Topp and Leone probability density function.
GTLPDF	= Compute the generalized Topp and Leone probability density function.
TOPPDF	= Compute the Topp and Leone probability density function.
TSPPDF	= Compute the two-sided power probability density function.
BETPDF	= Compute the beta probability density function.
TRIPDF	= Compute the triangular probability density function.
TRAPDF	= Compute the trapezoid probability density function.
UNIPDF	= Compute the uniform probability density function.
POWPDF	= Compute the power probability density function.
JSBPDF	= Compute the Johnson SB probability density function.

Kumaraswamy (1980), "A Generalized Probability Density Function for Double-Bounded Random Processes", Journal of Hydrology, 46: 79-88.

Distributional Modeling

2007/11

 
CASE ASIS
LABEL CASE ASIS
TITLE CASE ASIS
TITLE OFFSET 2
.
MULTIPLOT 3 3
MULTIPLOT CORNER COORDINATES 0 0 100 95
MULTIPLOT SCALE FACTOR 3
.
LET ALPHA = 2
LET BETA  = 3
TITLE Alpha = ^alpha, Beta = ^beta
PLOT KUMCDF(X,ALPHA,BETA) FOR X = 0  0.01  1
.
LET ALPHA = 1.5
LET BETA  = 6
TITLE Alpha = ^alpha, Beta = ^beta
PLOT KUMCDF(X,ALPHA,BETA) FOR X = 0  0.01  1
.
LET ALPHA = 1.5
LET BETA  = 2
TITLE Alpha = ^alpha, Beta = ^beta
PLOT KUMCDF(X,ALPHA,BETA) FOR X = 0  0.01  1
.
LET ALPHA = 1.5
LET BETA  = 1
TITLE Alpha = ^alpha, Beta = ^beta
PLOT KUMCDF(X,ALPHA,BETA) FOR X = 0  0.01  1
.
LET ALPHA = 0.5
LET BETA  = 2
TITLE Alpha = ^alpha, Beta = ^beta
PLOT KUMCDF(X,ALPHA,BETA) FOR X = 0  0.01  1
.
LET ALPHA = 0.5
LET BETA  = 1
TITLE Alpha = ^alpha, Beta = ^beta
PLOT KUMCDF(X,ALPHA,BETA) FOR X = 0  0.01  1
.
LET ALPHA = 0.5
LET BETA  = 0.75
TITLE Alpha = ^alpha, Beta = ^beta
PLOT KUMCDF(X,ALPHA,BETA) FOR X = 0  0.01  1
.
LET ALPHA = 0.5
LET BETA  = 0.25
TITLE Alpha = ^alpha, Beta = ^beta
PLOT KUMCDF(X,ALPHA,BETA) FOR X = 0  0.01  1
.
LET ALPHA = 1
LET BETA  = 1
TITLE Alpha = ^alpha, Beta = ^beta
PLOT KUMCDF(X,ALPHA,BETA) FOR X = 0  0.01  1
.
END OF MULTIPLOT
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JUSTIFICATION CENTER
MOVE 50 97
TEXT Kumaraswamy Cumulative Distribution Functions