Cumulative Distribution Function

Gamma distribution cumulative distribution function.

The cumulative distribution function for a gamma random variable is

upper F left-parenthesis x semicolon alpha comma beta right-parenthesis equals integral Subscript 0 Superscript x Baseline f left-parenthesis u semicolon alpha comma beta right-parenthesis d u equals StartFraction gamma left-parenthesis alpha comma beta x right-parenthesis Over normal upper Gamma left-parenthesis alpha right-parenthesis EndFraction

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

Usage

var cdf = require( '@stdlib/math/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/math/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 ) );
}