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

Standard library statistical functions.

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# Variance

> [Erlang][erlang-distribution] distribution [variance][variance].

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The [variance][variance] for an [Erlang][erlang-distribution] random variable is

<!-- <equation class="equation" label="eq:erlang_variance" align="center" raw="\operatorname{Var}\left( X \right) = \frac{k}{\lambda^{2}}" alt="Variance for an Erlang distribution."> -->

<div class="equation" align="center" data-raw-text="\operatorname{Var}\left( X \right) = \frac{k}{\lambda^{2}}" data-equation="eq:erlang_variance">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@51534079fef45e990850102147e8945fb023d1d0/lib/node_modules/@stdlib/stats/base/dists/erlang/variance/docs/img/equation_erlang_variance.svg" alt="Variance for an Erlang distribution.">
    <br>
</div>

<!-- </equation> -->

where `k` is the shape parameter and `λ` is the rate parameter.

</section>

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## Usage

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

#### variance( k, lambda )

Returns the [variance][variance] of an [Erlang][erlang-distribution] distribution with parameters `k` (shape parameter) and `lambda` (rate parameter).

```javascript
var v = variance( 1, 1.0 );
// returns 1.0

v = variance( 4, 12.0 );
// returns ~0.028

v = variance( 8, 2.0 );
// returns 2.0
```

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

```javascript
var v = variance( NaN, 2.0 );
// returns NaN

v = variance( 2.0, NaN );
// returns NaN
```

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

```javascript
var v = variance( 1.8, 1.0 );
// returns NaN

v = variance( -1.0, 1.0 );
// returns NaN
```

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

```javascript
var v = variance( 2, 0.0 );
// returns NaN

v = variance( 2, -1.0 );
// returns NaN
```

</section>

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</section>

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## Examples

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

```javascript
var randu = require( '@stdlib/random/base/randu' );
var round = require( '@stdlib/math/base/special/round' );
var EPS = require( '@stdlib/constants/float64/eps' );
var variance = require( '@stdlib/stats/base/dists/erlang/variance' );

var lambda;
var k;
var v;
var i;

for ( i = 0; i < 10; i++ ) {
    k = round( randu()*10.0 );
    lambda = ( randu()*10.0 ) + EPS;
    v = variance( k, lambda );
    console.log( 'k: %d, λ: %d, Var(X;k,λ): %d', k.toFixed( 4 ), lambda.toFixed( 4 ), v.toFixed( 4 ) );
}
```

</section>

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</section>

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</section>

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<section class="links">

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

[variance]: https://en.wikipedia.org/wiki/Variance

</section>

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