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# Quantile Function

> [Poisson][poisson-distribution] distribution [quantile function][quantile-function].

<section class="intro">

The [quantile function][quantile-function] for a [Poisson][poisson-distribution] random variable returns for `0 <= p <= 1` the smallest nonnegative integer for which

<!-- <equation class="equation" label="eq:poisson_condition" align="center" raw="F(x;\lambda) \ge p" alt="Quantile condition."> -->

<div class="equation" align="center" data-raw-text="F(x;\lambda) \ge p" data-equation="eq:poisson_condition">
    <img src="https://cdn.rawgit.com/stdlib-js/stdlib/7e0a95722efd9c771b129597380c63dc6715508b/lib/node_modules/@stdlib/stats/base/dists/poisson/quantile/docs/img/equation_poisson_condition.svg" alt="Quantile condition.">
    <br>
</div>

<!-- </equation> -->

where `F` is the cumulative distribution function (CDF) of a Poisson distribution with mean parameter `lambda > 0`.

</section>

<!-- /.intro -->

<section class="usage">

## Usage

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

#### quantile( p, lambda )

Evaluates the [quantile function][quantile-function] for a [Poisson][poisson-distribution] distribution with mean parameter `lambda` at a probability `p`.

```javascript
var y = quantile( 0.5, 2.0 );
// returns 2

y = quantile( 0.9, 4.0 );
// returns 7

y = quantile( 0.1, 200.0 );
// returns 182
```

If provided an input probability `p` outside the interval `[0,1]`, the function returns `NaN`.

```javascript
var y = quantile( 1.9, 0.5 );
// returns NaN

y = quantile( -0.1, 0.5 );
// returns NaN
```

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

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

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

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

```javascript
var y = quantile( 0.4, -1.0 );
// returns NaN
```

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

```javascript
var y = quantile( 0.1, 0.0 );
// returns 0.0

y = quantile( 0.9, 0.0 );
// returns 0.0
```

#### quantile.factory( lambda )

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

```javascript
var myquantile = quantile.factory( 5.0 );
var y = myquantile( 0.4 );
// returns 4

y = myquantile( 0.8 );
// returns 7

y = myquantile( 1.0 );
// returns Infinity
```

</section>

<!-- /.usage -->

<section class="examples">

## Examples

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

```javascript
var randu = require( '@stdlib/random/base/randu' );
var quantile = require( '@stdlib/stats/base/dists/poisson/quantile' );

var lambda;
var p;
var y;
var i;

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

</section>

<!-- /.examples -->

<section class="links">

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

[quantile-function]: https://en.wikipedia.org/wiki/Quantile_function

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

</section>

<!-- /.links -->
