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feat: add docs, benchmarks and examples
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‎lib/node_modules/@stdlib/stats/base/dists/bradford/pdf/README.md‎

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<!--
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@license Apache-2.0
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Copyright (c) 2025 The Stdlib Authors.
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Licensed under the Apache License, Version 2.0 (the "License");
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you may not use this file except in compliance with the License.
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You may obtain a copy of the License at
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http://www.apache.org/licenses/LICENSE-2.0
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Unless required by applicable law or agreed to in writing, software
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distributed under the License is distributed on an "AS IS" BASIS,
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WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
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See the License for the specific language governing permissions and
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limitations under the License.
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-->
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# Probability Density Function
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> [Bradford][bradford-distribution] distribution [probability density function][pdf] (PDF).
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<section class="intro">
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The [probability density function][pdf] (PDF) for a [bradford][bradford-distribution] random variable is
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<!-- <equation class="equation" label="eq:bradford_pdf" align="center" raw="f(x;c)=\frac{c}{(\ln(1+c)) (1 + cx)}" alt="Probability density function (PDF) for a bradford distribution."> -->
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```math
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f(x;c)=\frac{c}{(\ln(1+c)) (1 + cx)}
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```
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<!-- <div class="equation" align="center" data-raw-text="f(x;c)=\frac{c}{(\ln(1+c)) (1 + cx)}" data-equation="eq:bradford_pdf">
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<img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@51534079fef45e990850102147e8945fb023d1d0/lib/node_modules/@stdlib/stats/base/dists/bradford/pdf/docs/img/equation_bradford_pdf.svg" alt="Probability density function (PDF) for a bradford distribution.">
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<br>
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</div> -->
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<!-- </equation> -->
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where `c` is the shape parameter of the distribution. The parameters must satisfy `0 <= x <= 1` and `c > 0`.
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</section>
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<!-- /.intro -->
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<section class="usage">
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## Usage
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```javascript
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var pdf = require( '@stdlib/stats/base/dists/bradford/pdf' );
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```
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#### pdf( x, c )
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Evaluates the [probability density function][pdf] (PDF) for a [bradford][bradford-distribution] distribution with shape parameter `c` at a value `x`.
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```javascript
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var y = pdf( 0.1, 0.1 );
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// returns ~1.039
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y = pdf( 0.5, 5.0 );
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// returns ~0.797
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y = pdf( 1.0, 10.0 );
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// returns ~0.379
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```
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If provided `NaN` as any argument, the function returns `NaN`.
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```javascript
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var y = pdf( NaN, 1.0 );
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// returns NaN
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y = pdf( 0.0, NaN );
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// returns NaN
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```
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If provided an input value `x` outside of the interval `[0,1]`, the function returns `NaN`.
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```javascript
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var y = pdf( 2.0, 1.0 );
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// returns NaN
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y = pdf( -0.5, 1.0 );
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// returns NaN
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```
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If provided a shape parameter `c <= 0`, the function returns `NaN`.
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```javascript
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var y = pdf( 0.0, 0.0 );
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// returns NaN
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y = pdf( 0.5, -1.0 );
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// returns NaN
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```
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#### pdf.factory( c )
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Returns a `function` for evaluating the [PDF][pdf] of a [bradford][bradford-distribution] distribution with shape parameter `c`.
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```javascript
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var myPDF = pdf.factory( 5.0 );
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var y = myPDF( 0.5 );
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// returns ~0.797
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y = myPDF( 1.0 );
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// returns ~0.465
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```
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</section>
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<!-- /.usage -->
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<section class="examples">
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## Examples
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<!-- eslint no-undef: "error" -->
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```javascript
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var randu = require( '@stdlib/random/base/randu' );
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var pdf = require( '@stdlib/stats/base/dists/bradford/pdf' );
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var x;
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var c;
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var y;
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var i;
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for ( i = 0; i < 25; i++ ) {
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x = randu();
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c = ( randu()*10.0 );
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y = pdf( x, c );
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console.log( 'x: %d, c: %d, f(x;c): %d', x.toFixed( 4 ), c.toFixed( 4 ), y.toFixed( 4 ) );
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}
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```
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</section>
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<!-- /.examples -->
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<!-- Section for related `stdlib` packages. Do not manually edit this section, as it is automatically populated. -->
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<section class="related">
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</section>
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<!-- /.related -->
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<!-- Section for all links. Make sure to keep an empty line after the `section` element and another before the `/section` close. -->
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<section class="links">
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[pdf]: https://en.wikipedia.org/wiki/Probability_density_function
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[bradford-distribution]: https://en.wikipedia.org/wiki/Bradford%27s_law
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</section>
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<!-- /.links -->

‎lib/node_modules/@stdlib/stats/base/dists/bradford/pdf/benchmark/benchmark.js‎

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/**
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* @license Apache-2.0
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*
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* Copyright (c) 2025 The Stdlib Authors.
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*
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* Licensed under the Apache License, Version 2.0 (the "License");
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* you may not use this file except in compliance with the License.
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* You may obtain a copy of the License at
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*
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* http://www.apache.org/licenses/LICENSE-2.0
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*
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* Unless required by applicable law or agreed to in writing, software
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* distributed under the License is distributed on an "AS IS" BASIS,
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* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
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* See the License for the specific language governing permissions and
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* limitations under the License.
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*/
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'use strict';
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// MODULES //
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var bench = require( '@stdlib/bench' );
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var Float64Array = require( '@stdlib/array/float64' );
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var uniform = require( '@stdlib/random/base/uniform' );
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var EPS = require( '@stdlib/constants/float64/eps' );
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var isnan = require( '@stdlib/math/base/assert/is-nan' );
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var pkg = require( './../package.json' ).name;
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var pdf = require( './../lib' );
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// MAIN //
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bench( pkg, function benchmark( b ) {
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var len;
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var x;
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var c;
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var y;
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var i;
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len = 100;
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x = new Float64Array( len );
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c = new Float64Array( len );
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for ( i = 0; i < len; i++ ) {
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x[ i ] = uniform( 0.0, 1.0 );
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c[ i ] = uniform( EPS, 10.0 );
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}
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b.tic();
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for ( i = 0; i < b.iterations; i++ ) {
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y = pdf( x[ i % len ], c[ i % len ] );
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if ( isnan( y ) ) {
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b.fail( 'should not return NaN' );
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}
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}
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b.toc();
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if ( isnan( y ) ) {
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b.fail( 'should not return NaN' );
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}
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b.pass( 'benchmark finished' );
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b.end();
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});

‎lib/node_modules/@stdlib/stats/base/dists/bradford/pdf/docs/img/equation_bradford_pdf.svg‎

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‎lib/node_modules/@stdlib/stats/base/dists/bradford/pdf/docs/repl.txt‎

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{{alias}}( x, c )
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Evaluates the probability density function (PDF) for a bradford distribution
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with shape parameter `c` at a value `x`.
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If provided `NaN` as any argument, the function returns `NaN`.
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If `x < 0` or `x > 1`, the function returns `NaN`.
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If provided `c <= 0`, the function returns `NaN`.
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Parameters
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----------
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x: number
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Input value.
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c: number
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Shape parameter.
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Returns
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-------
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out: number
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Evaluated PDF.
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Examples
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--------
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> var y = {{alias}}( 0.1, 0.1 )
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~1.039
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> y = {{alias}}( 0.5, 5.0 )
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~0.797
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> y = {{alias}}( 1.0, 10.0 )
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~0.379
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> y = {{alias}}( 0.5, 0.0 )
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NaN
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> y = {{alias}}( 2.0, 0.5 )
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NaN
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> y = {{alias}}( -1.0, 0.5 )
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NaN
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> y = {{alias}}( NaN, 1.0 )
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NaN
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> y = {{alias}}( 1.0, NaN )
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NaN
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{{alias}}.factory( c )
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Returns a function for evaluating the probability density function (PDF) of
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a bradford distribution with shape parameter `c`.
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Parameters
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----------
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c: number
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Shape parameter.
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Returns
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-------
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pdf: Function
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Probability density function (PDF).
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Examples
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--------
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> var myPDF = {{alias}}.factory( 5.0 );
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> var y = myPDF( 0.5 )
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~0.797
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> y = myPDF( 1.0 )
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~0.465
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See Also
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--------
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