NCTPDF

Updated 2024-03-12 20:45:54.030000

Syntax

SELECT [westclintech].[wct].[NCTPDF] (
  <@X, float,>
 ,<@DF, float,>
 ,<@Lambda, float,>)

Description

Use the scalar function NCTPDF to calculate the probability density function of the non-central T distribution.

The formula for the probability density function is:

f(x)=  \begin{cases} \frac{\nu}{x} \left \{ F_{\nu+2,\mu} \left (x\sqrt{1+\frac{2}{\nu}} \right ) - F_{\nu,\mu}(x)\right \}, &\mbox{if } x\neq 0; \\\\ \frac{\Gamma(\frac{\nu+1}{2})}{\sqrt{\pi\nu} \Gamma(\frac{\nu}{2})} \exp\left (-\frac{\mu^2}{2}\right), &\mbox{if } x=0. \end{cases}

Where Fν,μ(x) is the noncentral T cumulative distribution function.

Arguments

@X

is the variable to be evaluated. @X is an expression of type float or of a type that implicitly converts to float.

@Lambda

is the non-centrality parameter. @Lambda is an expression of type float or of a type that implicitly converts to float.

@DF

degrees of freedom. @DF is an expression of type float or of a type that implicitly converts to float.

Return Type

float

Remarks

@X must be greater than zero (@X > 0).

@DF must be greater than zero (@DF > 0).

@Lambda must be greater than or equal to zero (@Lambda > 0).

Examples

Calculate the probability density function:

SELECT wct.NCTPDF(1, 3, 3);

This produces the following result.

{"columns":[{"field":"column 1","headerClass":"ag-right-aligned-header","cellClass":"ag-right-aligned-cell"}],"rows":[{"column 1":"0.0700404713849545"}]}

You can use the SeriesFloat function from the XLeratorDB/math library to generate a dataset which can be pasted into EXCEL to generate a graph of the probability density function.

SELECT SeriesValue,

       wct.NCTPDF(SeriesValue, 1, 1) as [f(x,1,1)],

       wct.NCTPDF(SeriesValue, 3, 1) as [f(x,3,1)],

       wct.NCTPDF(SeriesValue, 5, 1) as [f(x,5,1)],

       wct.NCTPDF(SeriesValue, 1, 3) as [f(x,1,3)],

       wct.NCTPDF(SeriesValue, 1, 5) as [f(x,1,5)]

FROM wct.SeriesFloat(.1, 10, .1, NULL, NULL);

This is an EXCEL-generated graph of the results.