npm package: @stdlib/stats-base-dists-poisson-cdf
Description: Poisson distribution cumulative distribution function (CDF).
Author: The Stdlib Authors
Latest version: 0.3.1

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# Cumulative Distribution 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 [cumulative distribution function][cdf].

<section class="intro">

The [cumulative distribution function][cdf] for a [Poisson][poisson-distribution] random variable is

<!-- <equation class="equation" label="eq:poisson_cdf" align="center" raw="F(x;\lambda) = \begin{cases} 0 & \text{ for } x \le 0 \\ e^{-\lambda} \sum_{i=0}^{\lfloor x\rfloor} \frac{\lambda^i}{i!} & \text{ for } x > 0 \end{cases}" alt="Cumulative distribution function for a Poisson distribution."> -->

<div class="equation" align="center" data-raw-text="F(x;\lambda) = \begin{cases} 0 &amp; \text{ for } x \le 0 \\ e^{-\lambda} \sum_{i=0}^{\lfloor x\rfloor} \frac{\lambda^i}{i!} &amp; \text{ for } x &gt; 0 \end{cases}" data-equation="eq:poisson_cdf">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@51534079fef45e990850102147e8945fb023d1d0/lib/node_modules/@stdlib/stats/base/dists/poisson/cdf/docs/img/equation_poisson_cdf.svg" alt="Cumulative distribution function for a Poisson distribution.">
    <br>
</div>

<!-- </equation> -->

where `lambda` is the mean parameter. Internally, the module evaluates the CDF by evaluating the upper regularized gamma function at input values `lambda` and `floor( x ) + 1`.

</section>

<!-- /.intro -->

<section class="installation">

## Installation

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

</section>

<section class="usage">

## Usage

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

#### cdf( x, lambda )

Evaluates the [cumulative distribution function][cdf] for a [Poisson][poisson-distribution] distribution with mean parameter `lambda`.

```javascript
var y = cdf( 2.0, 0.5 );
// returns ~0.986

y = cdf( 2.0, 10.0 );
// returns ~0.003

y = cdf( -1.0, 4.0 );
// returns 0.0
```

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

```javascript
var y = cdf( NaN, 1.0 );
// returns NaN

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

If provided `lambda < 0`, the function returns `NaN`.

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

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

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

y = cdf( 0.0, 0.0 );
// returns 1.0

y = cdf( 10.0, 0.0 );
// returns 1.0
```

#### cdf.factory( lambda )

Returns a function for evaluating the [cumulative distribution function][cdf] of a [Poisson][poisson-distribution] distribution with mean parameter `lambda`.

```javascript
var mycdf = cdf.factory( 5.0 );
var y = mycdf( 3.0 );
// returns ~0.265

y = mycdf( 8.0 );
// returns ~0.932
```

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

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

logEachMap( 'x: %0.4f, λ: %0.4f, F(x;λ): %0.4f', x, lambda, cdf );
```

</section>

<!-- /.examples -->

* * *

<section class="c">

## C APIs

<section class="intro">

</section>

<!-- /.intro -->

<section class="usage">

### Usage

```c
#include "stdlib/stats/base/dists/poisson/cdf.h"
```

#### stdlib_base_dists_poisson_cdf( x, lambda )

Evaluates the [cumulative distribution function][cdf] for a [Poisson][poisson-distribution] distribution with mean parameter `lambda`.

```c
double out = stdlib_base_dists_poisson_cdf( 2.0, 0.5 );
// returns ~0.986

out = stdlib_base_dists_poisson_cdf( 2.0, 10.0 );
// returns ~0.003
```

The function accepts the following arguments:

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

```c
double stdlib_base_dists_poisson_cdf( const double x, const double lambda );
```

</section>

<!-- /.usage -->

<section class="examples">

### Examples

```c
#include "stdlib/stats/base/dists/poisson/cdf.h"
#include <stdlib.h>
#include <stdio.h>

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

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

    for ( i = 0; i < 25; i++ ) {
        x = random_uniform( 0.0, 10.0 );
        lambda = random_uniform( 0.1, 20.0 );
        y = stdlib_base_dists_poisson_cdf( x, lambda );
        printf( "x: %lf, λ: %lf, F(x;λ): %lf\n", x, lambda, y );
    }
}
```

</section>

<!-- /.examples -->

</section>

<!-- /.c -->

<!-- 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].

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---

## License

See [LICENSE][stdlib-license].


## Copyright

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

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[cdf]: https://en.wikipedia.org/wiki/Cumulative_distribution_function

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

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

</section>

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