T_DIST

Updated 2023-11-05 12:18:38.057000

Syntax

SELECT [westclintech].[wct].[T_DIST](
  <@X, float,>
 ,<@df, float,>
 ,<@Cumulative, bit,>)

Description

Use the scalar function T_DIST to calculate the probability density function and the left-tailed cumulative probability distribution of Student's t-distribution.

Arguments

@df

the number of degrees of freedom. @df must of a type float or of a type that intrinsically converts to float.

@X

the value of interest to be evaluated. @X must be of a type float or of type that intrinsically converts to float.

@Cumulative

A bit value indicating whether the probability density function ( 'False') or the cumulative distribution function ( 'True') should be returned.

Return Type

float

Remarks

0 < @df.

Examples

In this example we calculate the probability density function for x = 8 with 3 degrees of freedom.

SELECT wct.T_DIST(   8,      --@X

                     3,      --@df

                     'False' --@Cumulative       

                 ) as pdf;

This produces the following result.

{"columns":[{"field":"pdf","headerClass":"ag-right-aligned-header","cellClass":"ag-right-aligned-cell"}],"rows":[{"pdf":"0.000736906520946926"}]}

In this example we calculate the left-tailed cumulative distribution function for x = 60 with 1 degree of freedom.

SELECT wct.T_DIST(   60,    --@X

                     1,     --@df

                     'True' --@Cumulative       

                 ) as cdf;

This produces the following result.

{"columns":[{"field":"cdf","headerClass":"ag-right-aligned-header","cellClass":"ag-right-aligned-cell"}],"rows":[{"cdf":"0.994695326367377"}]}

The left-tailed cumulative distribution is closely related to the beta distribution.c

SELECT wct.T_DIST(x, df, 'True') as T_DIST,

       (1 - wct.BETA_DIST(x * x / (df + x * X), 0.5, df * 0.5, 'True', NULL, NULL)

                 ) * 0.5 as BETA_DIST

FROM

(

    VALUES

        (-2e-00, 5e-00)

) n (x, df);

This produces the following result.

{"columns":[{"field":"T_DIST","headerClass":"ag-right-aligned-header","cellClass":"ag-right-aligned-cell"},{"field":"BETA_DIST","headerClass":"ag-right-aligned-header","cellClass":"ag-right-aligned-cell"}],"rows":[{"T_DIST":"0.0509697394149292","BETA_DIST":"0.0509697394149292"}]}