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Cumulative Distribution Function

Gamma distribution cumulative distribution function.

The cumulative distribution function for a gamma random variable is

$$F(x;\alpha,\beta) = \int_0^x f(u;\alpha,\beta)\,du= \frac{\gamma(\alpha, \beta x)}{\Gamma(\alpha)}$$

where alpha is the shape parameter and beta is the rate parameter of the distribution. gamma is the lower incomplete gamma function.

Installation

npm install @stdlib/stats-base-dists-gamma-cdf

Alternatively,

To load the package in a website via a script tag without installation and bundlers, use the ES Module available on the esm branch (see README).
If you are using Deno, visit the deno branch (see README for usage intructions).
For use in Observable, or in browser/node environments, use the Universal Module Definition (UMD) build available on the umd branch (see README).

The branches.md file summarizes the available branches and displays a diagram illustrating their relationships.

To view installation and usage instructions specific to each branch build, be sure to explicitly navigate to the respective README files on each branch, as linked to above.

Usage

var cdf = require( '@stdlib/stats-base-dists-gamma-cdf' );

cdf( x, alpha, beta )

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

var y = cdf( 2.0, 1.0, 1.0 );
// returns ~0.865

y = cdf( 2.0, 3.0, 1.0 );
// returns ~0.323

y = cdf( -1.0, 2.0, 2.0 );
// returns 0.0

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

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

If provided alpha = 0, the function evaluates the CDF of a degenerate distribution centered at 0.

var y = cdf( 2.0, 0.0, 2.0 );
// returns 1.0

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

y = cdf( 0.0, 0.0, 2.0 );
// returns 1.0

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

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

cdf.factory( alpha, beta )

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

var mycdf = cdf.factory( 0.5, 0.1 );

var y = mycdf( 12.0 );
// returns ~0.879

y = mycdf( 8.0 );
// returns ~0.794

Examples

var randu = require( '@stdlib/random-base-randu' );
var cdf = require( '@stdlib/stats-base-dists-gamma-cdf' );

var alpha;
var beta;
var x;
var y;
var i;

for ( i = 0; i < 10; i++ ) {
    x = randu() * 3.0;
    alpha = randu() * 5.0;
    beta = randu() * 5.0;
    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 ) );
}

Notice

This package is part of stdlib, a standard library for JavaScript and Node.js, with an emphasis on numerical and scientific computing. The library provides a collection of robust, high performance libraries for mathematics, statistics, streams, utilities, and more.

For more information on the project, filing bug reports and feature requests, and guidance on how to develop stdlib, see the main project repository.

Community

License

See LICENSE.

stdlib-js/stats-base-dists-gamma-cdf