Probability Density Function

Beta prime distribution probability density function (PDF).

The probability density function (PDF) for a beta prime random variable is

f left-parenthesis x semicolon alpha comma beta right-parenthesis equals StartLayout Enlarged left-brace 1st Row 1st Column StartFraction normal upper Gamma left-parenthesis alpha plus beta right-parenthesis Over normal upper Gamma left-parenthesis alpha right-parenthesis plus normal upper Gamma left-parenthesis beta right-parenthesis EndFraction x Superscript alpha minus 1 Baseline left-parenthesis 1 plus x right-parenthesis Superscript negative alpha minus beta Baseline 2nd Column for x greater-than 0 2nd Row 1st Column 0 2nd Column otherwise EndLayout

where α > 0 is the first shape parameter and β > 0 is the second shape parameter.

Usage

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

pdf( x, alpha, beta )

Evaluates the probability density function (PDF) for a beta prime distribution with parameters alpha (first shape parameter) and beta (second shape parameter).

var y = pdf( 0.5, 0.5, 1.0 );
// returns ~0.385

y = pdf( 0.1, 1.0, 1.0 );
// returns ~0.826

y = pdf( 0.8, 4.0, 2.0 );
// returns ~0.301

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

var y = pdf( -0.1, 1.0, 1.0 );
// returns 0.0

y = pdf( 0.0, 1.0, 1.0 );
// 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 alpha <= 0, the function returns NaN.

var y = pdf( 0.5, 0.0, 1.0 );
// returns NaN

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

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

var y = pdf( 0.5, 1.0, 0.0 );
// returns NaN

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

pdf.factory( alpha, beta )

Returns a function for evaluating the PDF for a beta prime distribution with parameters alpha (first shape parameter) and beta (second shape parameter).

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

var y = mypdf( 0.8 );
// returns ~0.198

y = mypdf( 0.3 );
// returns ~0.447

Examples

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

var alpha;
var beta;
var x;
var y;
var i;

for ( i = 0; i < 10; i++ ) {
    x = randu();
    alpha = ( randu()*5.0 ) + EPS;
    beta = ( randu()*5.0 ) + EPS;
    y = pdf( x, alpha, beta );
    console.log( 'x: %d, α: %d, β: %d, f(x;α,β): %d', x.toFixed( 4 ), alpha.toFixed( 4 ), beta.toFixed( 4 ), y.toFixed( 4 ) );
}