Probability Density Function

Fréchet distribution probability density function.

The probability density function for a Fréchet random variable is

f left-parenthesis x semicolon mu comma beta right-parenthesis equals StartFraction alpha Over s EndFraction left-parenthesis StartFraction x minus m Over s EndFraction right-parenthesis Superscript negative 1 minus alpha Baseline e Superscript minus left-parenthesis StartFraction x minus m Over s EndFraction right-parenthesis Super Superscript negative alpha

where alpha > 0 is the shape, s > 0 the scale and m the location parameter.

Usage

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

pdf( x, alpha, s, m )

Evaluates the probability density function (PDF) for a Fréchet distribution with shape alpha, scale s, and location m at a value x.

var y = pdf( 10.0, 2.0, 3.0, 5.0 );
// returns ~0.698

y = pdf( -3.0, 1.0, 2.0, -4.0 );
// returns ~0.135

y = pdf( 0.0, 2.0, 1.0, -1.0 );
// returns ~0.368

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

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

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

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

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

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

var y = pdf( 2.0, -0.1, 1.0, 1.0 );
// returns NaN

y = pdf( 2.0, 0.0, 1.0, 1.0 );
// returns NaN

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

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

y = pdf( 2.0, 1.0, 0.0, 1.0 );
// returns NaN

pdf.factory( alpha, s, m )

Returns a function for evaluating the probability density function of a Fréchet distribution with shape alpha, scale s, and location m.

var mypdf = pdf.factory( 3.0, 3.0, 5.0 );

var y = mypdf( 10.0 );
// returns ~0.806

y = mypdf( 7.0 );
// returns ~0.034

Examples

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

var alpha;
var m;
var s;
var x;
var y;
var i;

for ( i = 0; i < 100; i++ ) {
    alpha = randu() * 10.0;
    x = randu() * 10.0;
    s = randu() * 10.0;
    m = randu() * 10.0;
    y = pdf( x, alpha, s, m );
    console.log( 'x: %d, α: %d, s: %d, m: %d, f(x;α,s,m): %d', x.toFixed( 4 ), alpha.toFixed( 4 ), s.toFixed( 4 ), m.toFixed( 4 ), y.toFixed( 4 ) );
}