non_central_t_pdf

Evaluates the noncentral Student's t probability density function.

Synopsis

#include <imsls.h>

float imsls_f_non_central_t_pdf (floatt, floatdf, floatdelta)

The type doublefunction is imsls_d_non_central_t_pdf.

Required Arguments

floatt (Input)
Argument for which the noncentral Student’s t probability density function is to be evaluated.

floatdf (Input)
Number of degrees of freedom of the noncentral Student’s t distribution. df must be greater than 0.

floatdelta (Input)
Noncentrality parameter.

Return Value

The probability density associated with a noncentral Student’s t random variable with value t.

Description

If w is a normally distributed random variable with unit variance and mean δ and u is a chi-square random variable with ν degrees of freedom that is statistically independent of w, then

 

is a noncentral t-distributed random variable with ν degrees of freedom and noncentrality parameter δ, that is,with ν = df, and δ = delta. The probability density function for the noncentral t-distribution is:

 

where

 

and t = t.

For δ = 0, the PDF reduces to the (central) Student’s tPDF:

 

and, for t = 0, the PDF becomes:

 

Example

This example calculates the noncentral Student’s tPDF for a distribution with 2 degrees of freedom and noncentrality parameter δ = 10.

 

#include <imsls.h>

#include <stdio.h>

 

int main()

{

    int i;

    float t[] = {-.5, 1.5, 3.5, 7.5, 51.5, 99.5};

    float df = 2.0, delta =10.0, pdfv;

 

    printf ("\n\n df: %4.0f;  delta: %4.0f\n\n", df, delta);

    printf ("    t       pdf(t)\n");

 

    for (i=0; i<6; i++) {

        pdfv = imsls_f_non_central_t_pdf(t[i], df, delta);

        printf (" %5.1f  %12.4e\n",t[i], pdfv);

    }

}

 

Output

 

df: 2; delta: 10

 

t pdf(t)

-0.5 1.6399e-024

1.5 7.4417e-010

3.5 2.8972e-003

7.5 7.8853e-002

51.5 1.4215e-003

99.5 2.0290e-004