Cumulative Distribution Function

Inverse Gamma distribution cumulative distribution function.

The cumulative distribution function for an inverse gamma random variable is

upper F left-parenthesis x semicolon alpha comma beta right-parenthesis equals StartStartFraction normal upper Gamma left-parenthesis alpha comma StartFraction beta Over x EndFraction right-parenthesis OverOver normal upper Gamma left-parenthesis alpha right-parenthesis EndEndFraction equals upper Q left-parenthesis StartFraction beta Over x EndFraction comma alpha right-parenthesis

where alpha > 0 is the shape parameter and beta > 0 is the scale parameter. Q is the upper regularized incomplete gamma function.

Usage

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

cdf( x, alpha, beta )

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

var y = cdf( 2.0, 1.0, 1.0 );
// returns ~0.607

y = cdf( 2.0, 3.0, 1.0 );
// returns ~0.986

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 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 an inverse 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 ~0.683

y = mycdf( 8.0 );
// returns ~0.617

Examples

var randu = require( '@stdlib/random/base/randu' );
var cdf = require( '@stdlib/math/base/dists/invgamma/cdf' );

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

for ( i = 0; i < 10; i++ ) {
    x = randu() * 2.0;
    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 ) );
}