npm package: @stdlib/stats-base-dists-poisson-pmf
Description: Poisson distribution probability mass function (PMF).
Author: The Stdlib Authors
Latest version: 0.2.3

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# Probability Mass Function

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

> [Poisson][poisson-distribution] distribution [probability mass function][pmf] (PMF).

<section class="intro">

The [probability mass function][pmf] (PMF) for a [Poisson][poisson-distribution] random variable is

<!-- <equation class="equation" label="eq:poisson_pmf" align="center" raw="f(x;\lambda)=P(X=x;\lambda)=\begin{cases} \tfrac{\lambda^x}{x!}e^{-\lambda} & \text{ for } x = 0,1,2,\ldots \\ 0 & \text{ otherwise} \end{cases}" alt="Probability mass function (PMF) for a Poisson distribution."> -->

<div class="equation" align="center" data-raw-text="f(x;\lambda)=P(X=x;\lambda)=\begin{cases} \tfrac{\lambda^x}{x!}e^{-\lambda} &amp; \text{ for } x = 0,1,2,\ldots \\ 0 &amp; \text{ otherwise} \end{cases}" data-equation="eq:poisson_pmf">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@51534079fef45e990850102147e8945fb023d1d0/lib/node_modules/@stdlib/stats/base/dists/poisson/pmf/docs/img/equation_poisson_pmf.svg" alt="Probability mass function (PMF) for a Poisson distribution.">
    <br>
</div>

<!-- </equation> -->

where `lambda > 0` is the mean parameter.

</section>

<!-- /.intro -->

<section class="installation">

## Installation

```bash
npm install @stdlib/stats-base-dists-poisson-pmf
```

</section>

<section class="usage">

## Usage

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

#### pmf( x, lambda )

Evaluates the [probability mass function][pmf] (PMF) of a [Poisson][poisson-distribution] distribution with mean parameter `lambda`.

```javascript
var y = pmf( 4.0, 3.0 );
// returns ~0.168

y = pmf( 1.0, 3.0 );
// returns ~0.149

y = pmf( -1.0, 2.0 );
// returns 0.0
```

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

```javascript
var y = pmf( NaN, 2.0 );
// returns NaN

y = pmf( 0.0, NaN );
// returns NaN
```

If provided a negative mean parameter `lambda`, the function returns `NaN`.

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

y = pmf( 4.0, -2.0 );
// returns NaN
```

If provided `lambda = 0`, the function evaluates the [PMF][pmf] of a [degenerate distribution][degenerate-distribution] centered at `0.0`.

```javascript
var y = pmf( 2.0, 0.0 );
// returns 0.0

y = pmf( 0.0, 0.0 );
// returns 1.0
```

#### pmf.factory( lambda )

Returns a function for evaluating the [probability mass function][pmf] (PMF) of a [Poisson][poisson-distribution] distribution with mean parameter `lambda`.

```javascript
var mypmf = pmf.factory( 1.0 );
var y = mypmf( 3.0 );
// returns ~0.061

y = mypmf( 1.0 );
// returns ~0.368
```

</section>

<!-- /.usage -->

<section class="examples">

## Examples

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

```javascript
var discreteUniform = require( '@stdlib/random-array-discrete-uniform' );
var uniform = require( '@stdlib/random-array-uniform' );
var logEachMap = require( '@stdlib/console-log-each-map' );
var pmf = require( '@stdlib/stats-base-dists-poisson-pmf' );

var opts = {
    'dtype': 'float64'
};
var x = discreteUniform( 10, 0.0, 10.0, opts );
var lambda = uniform( 10, 0.0, 10.0, opts );

logEachMap( 'x: %d, λ: %0.4f, P(X=x;λ): %0.4f', x, lambda, pmf );
```

</section>

<!-- /.examples -->

<!-- Section for related `stdlib` packages. Do not manually edit this section, as it is automatically populated. -->

<section class="related">

</section>

<!-- /.related -->

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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].

#### Community

[![Chat][chat-image]][chat-url]

---

## License

See [LICENSE][stdlib-license].


## Copyright

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

</section>

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

[poisson-distribution]: https://en.wikipedia.org/wiki/Poisson_distribution

[pmf]: https://en.wikipedia.org/wiki/Probability_mass_function

</section>

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