npm package: @stdlib/stats-base-dists-poisson-logpmf
Description: Natural logarithm of the probability mass function (PMF) for a Poisson distribution.
Author: The Stdlib Authors
Latest version: 0.3.1

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# Logarithm of Probability Mass Function

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> Evaluate the natural logarithm of the [probability mass function][pmf] (PMF) for a [Poisson][poisson-distribution] distribution.

<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/logpmf/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-logpmf
```

</section>

<section class="usage">

## Usage

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

#### logpmf( x, lambda )

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

```javascript
var y = logpmf( 4.0, 3.0 );
// returns ~-1.784

y = logpmf( 1.0, 3.0 );
// returns ~-1.901

y = logpmf( -1.0, 2.0 );
// returns -Infinity
```

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

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

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

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

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

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

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

```javascript
var y = logpmf( 2.0, 0.0 );
// returns -Infinity

y = logpmf( 0.0, 0.0 );
// returns 0.0
```

#### logpmf.factory( lambda )

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

```javascript
var mylogpmf = logpmf.factory( 1.0 );
var y = mylogpmf( 3.0 );
// returns ~-2.792

y = mylogpmf( 1.0 );
// returns ~-1.0
```

</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 logpmf = require( '@stdlib/stats-base-dists-poisson-logpmf' );

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, ln(P(X=x;λ)): %0.4f', x, lambda, logpmf );
```

</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/poisson/logpmf.h"
```

#### stdlib_base_dists_poisson_logpmf( x, lambda )

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

```c
double out = stdlib_base_dists_poisson_logpmf( 4.0, 3.0 );
// returns ~-1.784
```

The function accepts the following arguments:

-   **x**: `[in] double` input value.
-   **lambda**: `[in] double` mean parameter.

```c
double stdlib_base_dists_poisson_logpmf( const double x, 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/poisson/logpmf.h"
#include "stdlib/random/base/ceil.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 x;
    double y;
    int i;
    for ( i = 0; i < 25; i++ ) {
        x = stdlib_base_ceil( random_uniform( 0.0, 10.0 ) );
        lambda = random_uniform( 0.0, 10.0);
        y = stdlib_base_dists_poisson_logpmf( x, lambda );
        printf( "x: %lf, λ: %lf, ln(P(X=x;λ)): %lf\n", x, lambda, y );
    }
}
```

</section>

<!-- /.examples -->

</section>

<!-- /.c -->

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<section class="related">

</section>

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* * *

## 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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[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

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