Logarithm of Cumulative Distribution Function

Evaluate the natural logarithm of the cumulative distribution function (CDF) for a Student's t distribution.

The cumulative distribution function (CDF) for a t distribution random variable is

upper F left-parenthesis x semicolon nu right-parenthesis equals 1 minus one-half StartStartFraction upper B e t a left-parenthesis StartFraction nu Over nu plus x squared EndFraction semicolon StartFraction nu Over 2 EndFraction comma one-half right-parenthesis OverOver upper B e t a left-parenthesis StartFraction nu Over 2 EndFraction comma one-half right-parenthesis EndEndFraction

where v > 0 is the degrees of freedom. In the definition, Beta( x; a, b ) denotes the lower incomplete beta function and Beta( a, b ) the beta function.

Usage

var logcdf = require( '@stdlib/math/base/dists/t/logcdf' );

logcdf( x, v )

Evaluates the natural logarithm of the cumulative distribution function (CDF) for a Student's t distribution with degrees of freedom v.

var y = logcdf( 2.0, 0.1 );
// returns ~-0.493

y = logcdf( 1.0, 2.0 );
// returns ~-0.237

y = logcdf( -1.0, 4.0 );
// returns ~-1.677

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

var y = logcdf( NaN, 1.0 );
// returns NaN

y = logcdf( 0.0, NaN );
// returns NaN

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

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

y = logcdf( 2.0, 0.0 );
// returns NaN

logcdf.factory( v )

Returns a function for evaluating the natural logarithm of the CDF of a Student's t distribution with degrees of freedom v.

var mylogcdf = logcdf.factory( 0.5 );
var y = mylogcdf( 3.0 );
// returns ~-0.203

y = mylogcdf( 1.0 );
// returns ~-0.358

Examples

var randu = require( '@stdlib/random/base/randu' );
var logcdf = require( '@stdlib/math/base/dists/t/logcdf' );

var v;
var x;
var y;
var i;

for ( i = 0; i < 10; i++ ) {
    x = (randu() * 6.0) - 3.0;
    v = randu() * 10.0;
    y = logcdf( x, v );
    console.log( 'x: %d, v: %d, ln(F(x;v)): %d', x.toFixed( 4 ), v.toFixed( 4 ), y.toFixed( 4 ) );
}