# Cumulative Distribution Function

Beta distribution cumulative distribution function.

The cumulative distribution function for a beta random variable is

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.

## Usage

var cdf = require( '@stdlib/math/base/dists/beta/cdf' );


#### cdf( x, alpha, beta )

Evaluates the cumulative distribution function (CDF) for a beta distribution with parameters alpha (first shape parameter) and beta (second shape parameter).

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

y = cdf( 0.5, 2.0, 4.0 );
// returns ~0.813

y = cdf( 0.2, 2.0, 2.0 );
// returns ~0.104

y = cdf( 0.8, 4.0, 4.0 );
// returns ~0.967

y = cdf( -0.5, 4.0, 2.0 );
// returns 0.0

y = cdf( -Infinity, 4.0, 2.0 );
// returns 0.0

y = cdf( 1.5, 4.0, 2.0 );
// returns 1.0

y = cdf( +Infinity, 4.0, 2.0 );
// returns 1.0


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

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

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

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


If provided alpha <= 0, the function returns NaN.

var y = cdf( 2.0, -1.0, 0.5 );
// returns NaN

y = cdf( 2.0, 0.0, 0.5 );
// returns NaN


If provided beta <= 0, the function returns NaN.

var y = cdf( 2.0, 0.5, -1.0 );
// returns NaN

y = cdf( 2.0, 0.5, 0.0 );
// returns NaN


#### cdf.factory( alpha, beta )

Returns a function for evaluating the cumulative distribution function for a beta distribution with parameters alpha (first shape parameter) and beta (second shape parameter).

var mycdf = cdf.factory( 0.5, 0.5 );

var y = mycdf( 0.8 );
// returns ~0.705

y = mycdf( 0.3 );
// returns ~0.369


## Examples

var randu = require( '@stdlib/random/base/randu' );
var EPS = require( '@stdlib/constants/math/float64-eps' );
var cdf = require( '@stdlib/math/base/dists/beta/cdf' );

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 = cdf( x, alpha, beta );
console.log( 'x: %d, α: %d, β: %d, F(x;α,β): %d', x.toFixed( 4 ), alpha.toFixed( 4 ), beta.toFixed( 4 ), y.toFixed( 4 ) );
}