Cumulative Distribution Function

Kumaraswamy's double bounded distribution cumulative distribution function.

The cumulative distribution function for a Kumaraswamy's double bounded random variable is

upper F left-parenthesis x semicolon a comma b right-parenthesis equals 1 minus left-parenthesis 1 minus x Superscript a Baseline right-parenthesis Superscript b

where a > 0 is the first shape parameter and b > 0 is the second shape parameter.

Usage

var cdf = require( '@stdlib/math/base/dists/kumaraswamy/cdf' );

cdf( x, a, b )

Evaluates the cumulative distribution function (CDF) for a Kumaraswamy's double bounded distribution with parameters a (first shape parameter) and b (second shape parameter).

var y = cdf( 0.5, 1.0, 1.0 );
// returns 0.5

y = cdf( 0.5, 2.0, 4.0 );
// returns ~0.684

y = cdf( 0.2, 2.0, 2.0 );
// returns ~0.078

y = cdf( 0.8, 4.0, 4.0 );
// returns ~0.878

y = cdf( -0.5, 4.0, 2.0 );
// returns 0.0

y = cdf( -Infinity, 4.0, 2.0 );
// returns 0.0

y = cdf( 1.5, 4.0, 2.0 );
// returns 1.0

y = cdf( +Infinity, 4.0, 2.0 );
// returns 1.0

If provided NaN as any argument, the function returns NaN.

var y = cdf( NaN, 1.0, 1.0 );
// returns NaN

y = cdf( 0.0, NaN, 1.0 );
// returns NaN

y = cdf( 0.0, 1.0, NaN );
// returns NaN

If provided a <= 0, the function returns NaN.

var y = cdf( 2.0, -1.0, 0.5 );
// returns NaN

y = cdf( 2.0, 0.0, 0.5 );
// returns NaN

If provided b <= 0, the function returns NaN.

var y = cdf( 2.0, 0.5, -1.0 );
// returns NaN

y = cdf( 2.0, 0.5, 0.0 );
// returns NaN

cdf.factory( a, b )

Returns a function for evaluating the cumulative distribution function for a Kumaraswamy's double bounded distribution with parameters a (first shape parameter) and b (second shape parameter).

var mycdf = cdf.factory( 0.5, 0.5 );

var y = mycdf( 0.8 );
// returns ~0.675

y = mycdf( 0.3 );
// returns ~0.327

Examples

var randu = require( '@stdlib/random/base/randu' );
var EPS = require( '@stdlib/constants/math/float64-eps' );
var cdf = require( '@stdlib/math/base/dists/kumaraswamy/cdf' );

var a;
var b;
var x;
var y;
var i;

for ( i = 0; i < 10; i++ ) {
    x = randu();
    a = ( randu()*5.0 ) + EPS;
    b = ( randu()*5.0 ) + EPS;
    y = cdf( x, a, b );
    console.log( 'x: %d, a: %d, b: %d, F(x;a,b): %d', x.toFixed( 4 ), a.toFixed( 4 ), b.toFixed( 4 ), y.toFixed( 4 ) );
}