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

Standard library statistical functions.

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# Cumulative Distribution Function

> [Erlang][erlang-distribution] distribution [cumulative distribution function][cdf].

<section class="intro">

The [cumulative distribution function][cdf] for a [Erlang][erlang-distribution] random variable is

<!-- <equation class="equation" label="eq:erlang_cdf" align="center" raw="F(x; k,\lambda) = 1 - \sum_{n=0}^{k-1}\frac{1}{n!}e^{-\lambda x}(\lambda x)^n" alt="Cumulative distribution function for a Erlang distribution."> -->

<div class="equation" align="center" data-raw-text="F(x; k,\lambda) = 1 - \sum_{n=0}^{k-1}\frac{1}{n!}e^{-\lambda x}(\lambda x)^n" data-equation="eq:erlang_cdf">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@51534079fef45e990850102147e8945fb023d1d0/lib/node_modules/@stdlib/stats/base/dists/erlang/cdf/docs/img/equation_erlang_cdf.svg" alt="Cumulative distribution function for a Erlang distribution.">
    <br>
</div>

<!-- </equation> -->

where `k` is the shape parameter and `lambda` is the rate parameter. The [Erlang][erlang-distribution] distribution is a special case of the gamma distribution, as `k` is constrained to the natural numbers.

</section>

<!-- /.intro -->

<section class="usage">

## Usage

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

#### cdf( x, k, lambda )

Evaluates the [cumulative distribution function][cdf] (CDF) for an [Erlang][erlang-distribution] distribution with parameters `k` (shape parameter) and `lambda` (rate parameter).

```javascript
var y = cdf( 2.0, 1, 1.0 );
// returns ~0.865

y = cdf( 2.0, 3, 1.0 );
// returns ~0.323

y = cdf( -1.0, 2, 2.0 );
// returns 0.0

y = cdf( -Infinity, 4, 2.0 );
// returns 0.0

y = cdf( +Infinity, 4, 2.0 );
// returns 1.0
```

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

```javascript
var y = cdf( NaN, 1, 1.0 );
// returns NaN

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

y = cdf( 0.0, 1, NaN );
// returns NaN
```

If not provided a nonnegative integer for `k`, the function returns `NaN`.

```javascript
var y = cdf( 2.0, -2, 0.5 );
// returns NaN

y = cdf( 2.0, 0.5, 0.5 );
// returns NaN
```

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

```javascript
var y = cdf( 2.0, 0.0, 2.0 );
// returns 1.0

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

y = cdf( 0.0, 0.0, 2.0 );
// returns 1.0
```

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

```javascript
var y = cdf( 2.0, 1, 0.0 );
// returns NaN

y = cdf( 2.0, 1, -5.0 );
// returns NaN
```

#### cdf.factory( k, lambda )

Returns a function for evaluating the [cumulative distribution function][cdf] for an [Erlang][erlang-distribution] distribution with parameters `k` (shape parameter) and `lambda` (rate parameter).

```javascript
var mycdf = cdf.factory( 2, 0.5 );

var y = mycdf( 6.0 );
// returns ~0.801

y = mycdf( 2.0 );
// returns ~0.264
```

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

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

for ( i = 0; i < 20; i++ ) {
    x = randu() * 10.0;
    k = round( randu() * 10.0 );
    lambda = randu() * 5.0;
    y = cdf( x, k, lambda );
    console.log( 'x: %d, k: %d, λ: %d, F(x;k,λ): %d', x.toFixed( 4 ), k, 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 -->

<!-- Section for all links. Make sure to keep an empty line after the `section` element and another before the `/section` close. -->

<section class="links">

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

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

[erlang-distribution]: https://en.wikipedia.org/wiki/Erlang_distribution

</section>

<!-- /.links -->