Cumulative Distribution Function

Beta prime distribution cumulative distribution function.

The cumulative distribution function for a beta prime random variable is

upper F left-parenthesis x semicolon alpha comma beta right-parenthesis equals StartLayout Enlarged left-brace 1st Row 1st Column upper I Subscript StartFraction x Over 1 plus x EndFraction Baseline left-parenthesis alpha comma beta right-parenthesis 2nd Column for x greater-than 0 2nd Row 1st Column 0 2nd Column otherwise EndLayout

where alpha > 0 is the first shape parameter, beta > 0 is the second shape parameter and I is the incomplete beta function.

Usage

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

cdf( x, alpha, beta )

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

var y = cdf( 0.5, 1.0, 1.0 );
// returns ~0.333

y = cdf( 0.5, 2.0, 4.0 );
// returns ~0.539

y = cdf( 0.2, 2.0, 2.0 );
// returns ~0.074

y = cdf( 0.8, 4.0, 4.0 );
// returns ~0.38

y = cdf( -0.5, 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

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 prime 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 ~0.465

y = mycdf( 0.3 );
// returns ~0.319

Examples

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