Probability Density Function

Kumaraswamy's double bounded distribution probability density function.

The probability density function (PDF) for a Kumaraswamy's double bounded random variable is

f left-parenthesis x semicolon a comma b right-parenthesis equals StartLayout Enlarged left-brace 1st Row 1st Column a b x Superscript a minus 1 Baseline left-parenthesis 1 minus x Superscript a Baseline right-parenthesis Superscript b minus 1 Baseline 2nd Column for x element-of left-parenthesis 0 comma 1 right-parenthesis 2nd Row 1st Column 0 2nd Column otherwise EndLayout

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

Usage

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

pdf( x, a, b )

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

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

y = pdf( 0.5, 2.0, 4.0 );
// returns ~1.688

y = pdf( 0.2, 2.0, 2.0 );
// returns ~0.768

y = pdf( 0.8, 4.0, 4.0 );
// returns ~1.686

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

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

y = pdf( 1.5, 4.0, 2.0 );
// returns 0.0

y = pdf( +Infinity, 4.0, 2.0 );
// returns 0.0

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

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

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

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

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

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

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

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

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

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

pdf.factory( a, b )

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

var mypdf = pdf.factory( 0.5, 0.5 );

var y = mypdf( 0.8 );
// returns ~0.86

y = mypdf( 0.3 );
// returns ~0.679

Examples

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

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 = pdf( 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 ) );
}