Probability Density Function

Student's t distribution probability density function (PDF).

The probability density function (PDF) for a t distribution random variable is

$$\frac{1} {\sqrt{\nu}\,B\left( \tfrac{1}{2}, \tfrac{\nu}{2} \right )} \left(1+\frac{x^2}{\nu} \right)^{-\frac{\nu+1}{2}}$$

where v > 0 is the degrees of freedom.

Usage

var pdf = require( '@stdlib/stats/base/dists/t/pdf' );

pdf( x, v )

Evaluates the probability density function (PDF) for a Student's t distribution with degrees of freedom v.

var y = pdf( 0.3, 4.0 );
// returns ~0.355

y = pdf( 2.0, 0.7 );
// returns ~0.058

y = pdf( -1.0, 0.5 );
// returns ~0.118

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

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

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

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

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

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

pdf.factory( v )

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

var mypdf = pdf.factory( 1.0 );
var y = mypdf( 3.0 );
// returns ~0.032

y = mypdf( 1.0 );
// returns ~0.159

Examples

var randu = require( '@stdlib/random/base/randu' );
var pdf = require( '@stdlib/stats/base/dists/t/pdf' );

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 = pdf( x, v );
    console.log( 'x: %d, v: %d, f(x;v): %d', x, v, y );
}

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C APIs

Usage

#include "stdlib/stats/base/dists/t/pdf.h"

stdlib_base_dists_t_pdf( x, v )

Evaluates the probability density function (PDF) for a Student's t distribution with degrees of freedom v.

double out = stdlib_base_dists_t_pdf( 0.3, 4.0 );
// returns ~0.355

The function accepts the following arguments:

• x: [in] double input value.
• v: [in] double degree of freedom.

double stdlib_base_dists_t_pdf( const double x, const double v );

Examples

#include "stdlib/stats/base/dists/t/pdf.h"
#include <stdlib.h>
#include <stdio.h>

static double random_uniform( const double min, const double max ) {
    double v = (double)rand() / ( (double)RAND_MAX + 1.0 );
    return min + ( v*(max-min) );
}

int main( void ) {
    double x;
    double v;
    double y;
    int i;

    for ( i = 0; i < 25; i++ ) {
        x = random_uniform( -10.0, 10.0 );
        v = random_uniform( 0.1, 10.0 );
        y = stdlib_base_dists_t_pdf( x, v );
        printf( "x: %lf, v: %lf, f(x;v): %lf\n", x, v, y );
    }
}