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Cumulative Distribution Function

Erlang distribution cumulative distribution function.

The cumulative distribution function for a Erlang random variable is

$$F(x; k,\lambda) = 1 - \sum_{n=0}^{k-1}\frac{1}{n!}e^{-\lambda x}(\lambda x)^n$$

where k is the shape parameter and lambda is the rate parameter. The Erlang distribution is a special case of the gamma distribution, as k is constrained to the natural numbers.

Installation

npm install @stdlib/stats-base-dists-erlang-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-erlang-cdf' );

cdf( x, k, lambda )

Evaluates the cumulative distribution function (CDF) for an Erlang distribution with parameters k (shape parameter) and lambda (rate parameter).

var y = cdf( 2.0, 1, 1.0 );
// returns ~0.865

y = cdf( 2.0, 3, 1.0 );
// returns ~0.323

y = cdf( -1.0, 2, 2.0 );
// returns 0.0

y = cdf( -Infinity, 4, 2.0 );
// returns 0.0

y = cdf( +Infinity, 4, 2.0 );
// returns 1.0

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

var y = cdf( NaN, 1, 1.0 );
// returns NaN

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

y = cdf( 0.0, 1, NaN );
// returns NaN

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

var y = cdf( 2.0, -2, 0.5 );
// returns NaN

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

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

var y = cdf( 2.0, 1, 0.0 );
// returns NaN

y = cdf( 2.0, 1, -5.0 );
// returns NaN

cdf.factory( k, lambda )

Returns a function for evaluating the cumulative distribution function for an Erlang distribution with parameters k (shape parameter) and lambda (rate parameter).

var mycdf = cdf.factory( 2, 0.5 );

var y = mycdf( 6.0 );
// returns ~0.801

y = mycdf( 2.0 );
// returns ~0.264

Examples

var randu = require( '@stdlib/random-base-randu' );
var round = require( '@stdlib/math-base-special-round' );
var cdf = require( '@stdlib/stats-base-dists-erlang-cdf' );

var lambda;
var k;
var x;
var y;
var i;

for ( i = 0; i < 20; i++ ) {
    x = randu() * 10.0;
    k = round( randu() * 10.0 );
    lambda = randu() * 5.0;
    y = cdf( x, k, lambda );
    console.log( 'x: %d, k: %d, λ: %d, F(x;k,λ): %d', x.toFixed( 4 ), k, lambda.toFixed( 4 ), y.toFixed( 4 ) );
}

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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

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License

See LICENSE.

Copyright

Copyright © 2016-2024. The Stdlib Authors.