@stdlib/stats/base/dists/poisson/cdf/README.md

Standard library statistical functions.

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# Cumulative Distribution Function

> [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="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 randu = require( '@stdlib/random/base/randu' );
var cdf = require( '@stdlib/stats/base/dists/poisson/cdf' );

var lambda;
var x;
var y;
var i;

for ( i = 0; i < 10; i++ ) {
    x = randu() * 10.0;
    lambda = randu() * 10.0;
    y = cdf( x, lambda );
    console.log( 'x: %d, λ: %d, F(x;λ): %d', x.toFixed( 4 ), lambda.toFixed( 4 ), y.toFixed( 4 ) );
}
```

</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="links">

[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>

<!-- /.links -->