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Probability Density Function

Wald distribution probability density function (PDF).

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

$$f(x;\mu,\lambda) = \sqrt{\frac{\lambda}{2\pi x^3}}\, e^{-\frac{\lambda(x-\mu)^2}{2\mu^2 x}}$$

where µ > 0 is the mean and λ > 0 is the shape parameter.

Installation

npm install @stdlib/stats-base-dists-wald-pdf

Alternatively,

• To load the package in a website via a script tag without installation and bundlers, use the ES Module available on the esm branch (see README).
• If you are using Deno, visit the deno branch (see README for usage intructions).
• For use in Observable, or in browser/node environments, use the Universal Module Definition (UMD) build available on the umd branch (see README).

The branches.md file summarizes the available branches and displays a diagram illustrating their relationships.

To view installation and usage instructions specific to each branch build, be sure to explicitly navigate to the respective README files on each branch, as linked to above.

Usage

var pdf = require( '@stdlib/stats-base-dists-wald-pdf' );

pdf( x, mu, lambda )

Evaluates the probability density function (PDF) for a Wald distribution with parameters mu (mean) and lambda (shape parameter).

var y = pdf( 2.0, 1.0, 1.0 );
// returns ~0.110

y = pdf( 0.5, 2.0, 3.0 );
// returns ~0.362

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

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

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

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

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

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

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

If provided lambda < 0, the function returns NaN.

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

If provided lambda = 0, the function evaluates the PDF of a degenerate distribution centered at mu.

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

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

If provided x <= 0, the function returns 0.0.

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

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

pdf.factory( mu, lambda )

Partially applies mu and lambda to create a reusable function for evaluating the PDF.

var mypdf = pdf.factory( 1.0, 1.0 );

var y = mypdf( 2.0 );
// returns ~0.110

y = mypdf( 0.5 );
// returns ~0.879

Examples

var uniform = require( '@stdlib/random-array-uniform' );
var logEachMap = require( '@stdlib/console-log-each-map' );
var EPS = require( '@stdlib/constants-float64-eps' );
var pdf = require( '@stdlib/stats-base-dists-wald-pdf' );

var opts = {
    'dtype': 'float64'
};
var x = uniform( 10, EPS, 10.0, opts );
var mu = uniform( 10, EPS, 10.0, opts );
var lambda = uniform( 10, EPS, 20.0, opts );

logEachMap( 'x: %0.4f, µ: %0.4f, λ: %0.4f, f(x;µ,λ): %0.4f', x, mu, lambda, pdf );

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C APIs

Usage

#include "stdlib/stats/base/dists/wald/pdf.h"

stdlib_base_dists_wald_pdf( x, mu, lambda )

Evaluates the probability density function (PDF) for a Wald distribution with parameters mu (mean) and lambda (shape parameter).

double y = stdlib_base_dists_wald_pdf( 2.0, 1.0, 1.0 );
// returns ~0.110

The function accepts the following arguments:

• x: [in] double input value.
• mu: [in] double mean.
• lambda: [in] double shape parameter.

double stdlib_base_dists_wald_pdf( const double x, const double mu, const double lambda );

Examples

#include "stdlib/stats/base/dists/wald/pdf.h"
#include "stdlib/constants/float64/eps.h"
#include <stdlib.h>
#include <stdio.h>

static double random_uniform( const double min, const double max ) {
    double v = (double)rand() / ( (double)RAND_MAX + 1.0 );
    return min + ( v*(max-min) );
}

int main( void ) {
    double lambda;
    double mu;
    double x;
    double y;
    int i;

    for ( i = 0; i < 10; i++ ) {
        x = random_uniform( 0.0, 10.0 );
        mu = random_uniform( STDLIB_CONSTANT_FLOAT64_EPS, 10.0 );
        lambda = random_uniform( STDLIB_CONSTANT_FLOAT64_EPS, 20.0 );
        y = stdlib_base_dists_wald_pdf( x, mu, lambda );
        printf( "x: %lf, µ: %lf, λ: %lf, f(x;µ,λ): %lf\n", x, mu, lambda, y );
    }
}

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Notice

This package is part of stdlib, a standard library for JavaScript and Node.js, with an emphasis on numerical and scientific computing. The library provides a collection of robust, high performance libraries for mathematics, statistics, streams, utilities, and more.

For more information on the project, filing bug reports and feature requests, and guidance on how to develop stdlib, see the main project repository.

Community

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License

See LICENSE.

Copyright

Copyright © 2016-2026. The Stdlib Authors.