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

Standard library statistical functions.

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# Cumulative Distribution Function

> [Normal][normal-distribution] distribution [cumulative distribution function][cdf].

<section class="intro">

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

<!-- <equation class="equation" label="eq:normal_cdf" align="center" raw="F(x;\mu,\sigma) = \frac{1}{2} \left[ 1 + \operatorname{erf}\left( \frac{x-\mu}{\sigma\sqrt{2}} \right) \right]" alt="Cumulative distribution function for a normal distribution."> -->

<div class="equation" align="center" data-raw-text="F(x;\mu,\sigma) = \frac{1}{2} \left[ 1 + \operatorname{erf}\left( \frac{x-\mu}{\sigma\sqrt{2}} \right) \right]" data-equation="eq:normal_cdf">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@51534079fef45e990850102147e8945fb023d1d0/lib/node_modules/@stdlib/stats/base/dists/normal/cdf/docs/img/equation_normal_cdf.svg" alt="Cumulative distribution function for a normal distribution.">
    <br>
</div>

<!-- </equation> -->

where `µ` is the mean and `σ` is the standard deviation.

</section>

<!-- /.intro -->

<section class="usage">

## Usage

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

#### cdf( x, mu, sigma )

Evaluates the [cumulative distribution function][cdf] (CDF) for a [normal][normal-distribution] distribution with parameters `mu` (mean) and `sigma` (standard deviation).

```javascript
var y = cdf( 2.0, 0.0, 1.0 );
// returns ~0.977

y = cdf( 0.0, 0.0, 1.0 );
// returns 0.5

y = cdf( -1.0, 4.0, 2.0 );
// returns ~0.006
```

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

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

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

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

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

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

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

```javascript
var y = cdf( 2.0, 8.0, 0.0 );
// returns 0.0

y = cdf( 8.0, 8.0, 0.0 );
// returns 1.0

y = cdf( 10.0, 8.0, 0.0 );
// returns 1.0
```

#### cdf.factory( mu, sigma )

Returns a function for evaluating the [cumulative distribution function][cdf] of a normal distribution with parameters `mu` and `sigma`.

```javascript
var mycdf = cdf.factory( 10.0, 2.0 );

var y = mycdf( 10.0 );
// returns 0.5

y = mycdf( 8.0 );
// returns ~0.159
```

</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/normal/cdf' );

var sigma;
var mu;
var x;
var y;
var i;

for ( i = 0; i < 10; i++ ) {
    x = randu() * 10.0;
    mu = (randu() * 10.0) - 5.0;
    sigma = randu() * 20.0;
    y = cdf( x, mu, sigma );
    console.log( 'x: %d, µ: %d, σ: %d, F(x;µ,σ): %d', x, mu, sigma, y );
}
```

</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

[normal-distribution]: https://en.wikipedia.org/wiki/Normal_distribution

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

</section>

<!-- /.links -->