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

Standard library statistical functions.

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# Probability Mass Function

> [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="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 randu = require( '@stdlib/random/base/randu' );
var round = require( '@stdlib/math/base/special/round' );
var pmf = require( '@stdlib/stats/base/dists/poisson/pmf' );

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

for ( i = 0; i < 10; i++ ) {
    x = round( randu() * 10.0 );
    lambda = randu() * 10.0;
    y = pmf( x, lambda );
    console.log( 'x: %d, λ: %d, P(X=x;λ): %d', x, 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">

[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>

<!-- /.links -->