npm package: @stdlib/stats-base-dists-wald-pdf
Description: Wald distribution probability density function (PDF).
Author: The Stdlib Authors
Latest version: 0.1.1

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# Probability Density Function

[![NPM version][npm-image]][npm-url] [![Build Status][test-image]][test-url] [![Coverage Status][coverage-image]][coverage-url] <!-- [![dependencies][dependencies-image]][dependencies-url] -->

> [Wald][wald-distribution] distribution probability density function (PDF).

<section class="intro">

The [probability density function][pdf] (PDF) for a [Wald][wald-distribution] random variable is

<!-- <equation class="equation" label="eq:wald_pdf" align="center" raw="f(x;\mu,\lambda) = \sqrt{\frac{\lambda}{2\pi x^3}}\, e^{-\frac{\lambda(x-\mu)^2}{2\mu^2 x}}" alt="Probability density function (PDF) for a Wald distribution."> -->

<div class="equation" align="center" data-raw-text="f(x;\mu,\lambda) = \sqrt{\frac{\lambda}{2\pi x^3}}\, e^{-\frac{\lambda(x-\mu)^2}{2\mu^2 x}}" data-equation="eq:wald_pdf">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@51534079fef45e990850102147e8945fb023d1d0/lib/node_modules/@stdlib/stats/base/dists/wald/pdf/docs/img/equation_wald_pdf.svg" alt="Probability density function (PDF) for a Wald distribution.">
    <br>
</div>

<!-- </equation> -->

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

</section>

<!-- /.intro -->

<section class="installation">

## Installation

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

</section>

<section class="usage">

## Usage

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

#### pdf( x, mu, lambda )

Evaluates the [probability density function][pdf] (PDF) for a [Wald][wald-distribution] distribution with parameters `mu` (mean) and `lambda` (shape parameter).

```javascript
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`.

```javascript
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`.

```javascript
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`.

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

If provided `lambda = 0`, the function evaluates the [PDF][pdf] of a [degenerate distribution][degenerate-distribution] centered at `mu`.

```javascript
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`.

```javascript
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.

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

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

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

</section>

<!-- /.usage -->

<section class="examples">

## Examples

<!-- eslint no-undef: "error" -->

```javascript
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 );
```

</section>

<!-- /.examples -->

<!-- C interface documentation. -->

* * *

<section class="c">

## C APIs

<!-- Section to include introductory text. Make sure to keep an empty line after the intro `section` element and another before the `/section` close. -->

<section class="intro">

</section>

<!-- /.intro -->

<!-- C usage documentation. -->

<section class="usage">

### Usage

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

#### stdlib_base_dists_wald_pdf( x, mu, lambda )

Evaluates the [probability density function][pdf] (PDF) for a [Wald][wald-distribution] distribution with parameters `mu` (mean) and `lambda` (shape parameter).

```c
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.

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

</section>

<!-- /.usage -->

<!-- C API usage notes. Make sure to keep an empty line after the `section` element and another before the `/section` close. -->

<section class="notes">

</section>

<!-- /.notes -->

<!-- C API usage examples. -->

<section class="examples">

### Examples

```c
#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 );
    }
}
```

</section>

<!-- /.examples -->

</section>

<!-- /.c -->

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<section class="references">

</section>

<!-- /.references -->

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<section class="related">

</section>

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<section class="main-repo" >

* * *

## Notice

This package is part of [stdlib][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][stdlib], see the main project [repository][stdlib].

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---

## License

See [LICENSE][stdlib-license].


## Copyright

Copyright &copy; 2016-2026. The Stdlib [Authors][stdlib-authors].

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[stdlib-license]: https://raw.githubusercontent.com/stdlib-js/stats-base-dists-wald-pdf/main/LICENSE

[pdf]: https://en.wikipedia.org/wiki/Probability_density_function

[wald-distribution]: https://en.wikipedia.org/wiki/Inverse_Gaussian_distribution

[degenerate-distribution]: https://en.wikipedia.org/wiki/Degenerate_distribution

</section>

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