time-to-botec

README.md (4783B)

---

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 # Logarithm of Cumulative Distribution Function
 
 > [Beta][beta-distribution] distribution logarithm of [cumulative distribution function][cdf].
 
 <section class="intro">
 
 The [cumulative distribution function][cdf] for a [beta][beta-distribution] random variable is
 
 <!-- <equation class="equation" label="eq:beta_cdf" align="center" raw="F(x;\alpha,\beta) = \frac{\operatorname{Beta}(x;\alpha,\beta)}{\operatorname{Beta}(\alpha,\beta)}" alt="Cumulative distribution function for a beta distribution."> -->
 
 <div class="equation" align="center" data-raw-text="F(x;\alpha,\beta) = \frac{\operatorname{Beta}(x;\alpha,\beta)}{\operatorname{Beta}(\alpha,\beta)}" data-equation="eq:beta_cdf">
     <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@51534079fef45e990850102147e8945fb023d1d0/lib/node_modules/@stdlib/stats/base/dists/beta/logcdf/docs/img/equation_beta_cdf.svg" alt="Cumulative distribution function for a beta distribution.">
     <br>
 </div>
 
 <!-- </equation> -->
 
 where `alpha > 0` is the first shape parameter and `beta > 0` is the second shape parameter. In the definition, `Beta( x; a, b )` denotes the lower incomplete beta function and `Beta( a, b )` the [beta function][beta-function].
 
 </section>
 
 <!-- /.intro -->
 
 <section class="usage">
 
 ## Usage
 
 ```javascript
 var logcdf = require( '@stdlib/stats/base/dists/beta/logcdf' );
 ```
 
 #### logcdf( x, alpha, beta )
 
 Evaluates the natural logarithm of the [cumulative distribution function][cdf] (CDF) for a [beta][beta-distribution] distribution with parameters `alpha` (first shape parameter) and `beta` (second shape parameter).
 
 ```javascript
 var y = logcdf( 0.5, 1.0, 1.0 );
 // returns ~-0.693
 
 y = logcdf( 0.5, 2.0, 4.0 );
 // returns ~-0.208
 
 y = logcdf( 0.2, 2.0, 2.0 );
 // returns ~-2.263
 
 y = logcdf( 0.8, 4.0, 4.0 );
 // returns ~-0.034
 
 y = logcdf( -0.5, 4.0, 2.0 );
 // returns -Infinity
 
 y = logcdf( -Infinity, 4.0, 2.0 );
 // returns -Infinity
 
 y = logcdf( 1.5, 4.0, 2.0 );
 // returns 0.0
 
 y = logcdf( +Infinity, 4.0, 2.0 );
 // returns 0.0
 ```
 
 If provided `NaN` as any argument, the function returns `NaN`.
 
 ```javascript
 var y = logcdf( NaN, 1.0, 1.0 );
 // returns NaN
 
 y = logcdf( 0.0, NaN, 1.0 );
 // returns NaN
 
 y = logcdf( 0.0, 1.0, NaN );
 // returns NaN
 ```
 
 If provided `alpha <= 0`, the function returns `NaN`.
 
 ```javascript
 var y = logcdf( 2.0, -1.0, 0.5 );
 // returns NaN
 
 y = logcdf( 2.0, 0.0, 0.5 );
 // returns NaN
 ```
 
 If provided `beta <= 0`, the function returns `NaN`.
 
 ```javascript
 var y = logcdf( 2.0, 0.5, -1.0 );
 // returns NaN
 
 y = logcdf( 2.0, 0.5, 0.0 );
 // returns NaN
 ```
 
 #### logcdf.factory( alpha, beta )
 
 Returns a function for evaluating the natural logarithm of the [cumulative distribution function][cdf] for a [beta][beta-distribution] distribution with parameters `alpha` (first shape parameter) and `beta` (second shape parameter).
 
 ```javascript
 var mylogcdf = logcdf.factory( 0.5, 0.5 );
 
 var y = mylogcdf( 0.8 );
 // returns ~-0.35
 
 y = mylogcdf( 0.3 );
 // returns ~-0.997
 ```
 
 </section>
 
 <!-- /.usage -->
 
 <section class="notes">
 
 ## Notes
 
 -   In virtually all cases, using the `logpdf` or `logcdf` functions is preferable to manually computing the logarithm of the `pdf` or `cdf`, respectively, since the latter is prone to overflow and underflow.
 
 </section>
 
 <!-- /.notes -->
 
 <section class="examples">
 
 ## Examples
 
 <!-- eslint no-undef: "error" -->
 
 ```javascript
 var randu = require( '@stdlib/random/base/randu' );
 var EPS = require( '@stdlib/constants/float64/eps' );
 var logcdf = require( '@stdlib/stats/base/dists/beta/logcdf' );
 
 var alpha;
 var beta;
 var x;
 var y;
 var i;
 
 for ( i = 0; i < 10; i++ ) {
     x = randu();
     alpha = ( randu()*5.0 ) + EPS;
     beta = ( randu()*5.0 ) + EPS;
     y = logcdf( x, alpha, beta );
     console.log( 'x: %d, α: %d, β: %d, ln(F(x;α,β)): %d', x.toFixed( 4 ), alpha.toFixed( 4 ), beta.toFixed( 4 ), y.toFixed( 4 ) );
 }
 ```
 
 </section>
 
 <!-- /.examples -->
 
 <section class="links">
 
 [beta-distribution]: https://en.wikipedia.org/wiki/Beta_distribution
 
 [beta-function]: https://en.wikipedia.org/wiki/Beta_function
 
 [cdf]: https://en.wikipedia.org/wiki/Cumulative_distribution_function
 
 </section>
 
 <!-- /.links -->