Probability Density Function

Arcsine distribution probability density function (PDF).

The probability density function (PDF) for an arcsine random variable is

f left-parenthesis x semicolon a comma b right-parenthesis equals StartLayout Enlarged left-brace 1st Row 1st Column StartFraction 1 Over pi StartRoot left-parenthesis x minus a right-parenthesis left-parenthesis b minus x right-parenthesis EndRoot EndFraction 2nd Column for x element-of left-bracket a comma b right-bracket 2nd Row 1st Column 0 2nd Column otherwise EndLayout

where a is the minimum support and b is the maximum support of the distribution. The parameters must satisfy a < b.

Usage

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

pdf( x, a, b )

Evaluates the probability density function (PDF) for an arcsine distribution with parameters a (minimum support) and b (maximum support).

var y = pdf( 2.0, 0.0, 4.0 );
// returns ~0.159

y = pdf( 5.0, 0.0, 4.0 );
// returns 0.0

y = pdf( 0.25, 0.0, 1.0 );
// returns ~0.735

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

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

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

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

If provided a >= b, the function returns NaN.

var y = pdf( 2.5, 3.0, 2.0 );
// returns NaN

y = pdf( 2.5, 3.0, 3.0 );
// returns NaN

pdf.factory( a, b )

Returns a function for evaluating the PDF of an arcsine distribution with parameters a (minimum support) and b (maximum support).

var myPDF = pdf.factory( 6.0, 7.0 );
var y = myPDF( 7.0 );
// returns Infinity

y = myPDF( 5.0 );
// returns 0.0

Examples

var randu = require( '@stdlib/random/base/randu' );
var pdf = require( '@stdlib/math/base/dists/arcsine/pdf' );

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

for ( i = 0; i < 25; i++ ) {
    x = ( randu()*20.0 )- 10.0;
    a = ( randu()*20.0 )- 20.0;
    b = a + ( randu()*40.0 );
    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 ) );
}