BETA_DIST

Updated 2024-02-13 20:41:09.970000

Syntax

SELECT [westclintech].[wct].[BETA_DIST](
  <@X, float,>
 ,<@Alpha, float,>
 ,<@Beta, float,>
 ,<@Cumulative, bit,>
 ,<@A, float,>
 ,<@B, float,>)

Description

Use the scalar function BETA_DIST to calculate the probability density function or cumulative distribution function of a beta distribution for x with shape parameters α and ß.

Arguments

@A

The lower bound of the interval for @X. @A must be of a type float or of a 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.

@B

The upper bound of the interval for @X. @B must be of a type float or of a type that intrinsically converts to float.

@Beta

The second shape parameter for the distribution. @Beta must be of a type float or of a type that intrinsically converts to float.

@Alpha

The first shape parameter for the distribution. @Alpha must be of a type float or of a type that intrinsically converts to float.

@X

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

Return Type

float

Remarks

@A ≤ @X ≤ @B.

@Alpha > 0.

@Beta > 0.

If @A is NULL then @A = 0.

If @B is NULL then @B = 1.

Examples

In this example we calculate the beta cumulative distribution function for x = 2, alpha = 8, beta = 10, with location parameters 8, 10.

SELECT wct.BETA_DIST(   2,      --@X

                        8,      --@Alpha

                        10,     --@Beta

                        'True', --@Cumulative

                        1,      --@A

                        3       --@B

                    ) as CDF;

This produces the following result.

{"columns":[{"field":"CDF","headerClass":"ag-right-aligned-header","cellClass":"ag-right-aligned-cell"}],"rows":[{"CDF":"0.685470581054688"}]}

Using the same data we calculate the beta probability density function.

SELECT wct.BETA_DIST(   2,       --@X

                        8,       --@Alpha

                        10,      --@Beta

                        'False', --@Cumulative

                        1,       --@A

                        3        --@B

                    ) as PDF;

This produces the following result.

{"columns":[{"field":"PDF","headerClass":"ag-right-aligned-header","cellClass":"ag-right-aligned-cell"}],"rows":[{"PDF":"1.4837646484375"}]}

This example demonstrates the recurrence relationship 26.5.10 from Abramowitz & Stegun .

DECLARE @x as float = 0.75;

DECLARE @Alpha as float = 4;

DECLARE @Beta as float = 10;

SELECT wct.BETA_DIST(@x, @alpha, @beta, 1, NULL, NULL) as cdf,

       @x * wct.BETA_DIST(@x, @alpha - 1, @beta, 1, NULL, NULL) + (1 - @x)

       * wct.BETA_DIST(@x, @alpha, @beta - 1, 1, NULL, NULL) as cdf2;

This produces the following result.

{"columns":[{"field":"cdf","headerClass":"ag-right-aligned-header","cellClass":"ag-right-aligned-cell"},{"field":"cdf2","headerClass":"ag-right-aligned-header","cellClass":"ag-right-aligned-cell"}],"rows":[{"cdf":"0.999873876571655","cdf2":"0.999873876571655"}]}

In this example we compare the results returned by the function to the results returned by integration.

DECLARE @x as float = 0.75
DECLARE @Alpha as float = 4
DECLARE @Beta as float = 10
SELECT
        wct.BETA_DIST(@x,@alpha,@beta,1,NULL,NULL) as cdf
       ,wct.QUADTS('SELECT POWER(@x,' + Cast(@alpha-1 as varchar(max)) + ')*POWER(1-@x,' + cast(@beta-1 as varchar(max)) + ')','@x',0,@x)/wct.BETA(@alpha,@beta) as [Integral];

This produces the following result.

{"columns":[{"field":"cdf","headerClass":"ag-right-aligned-header","cellClass":"ag-right-aligned-cell"},{"field":"Integral","headerClass":"ag-right-aligned-header","cellClass":"ag-right-aligned-cell"}],"rows":[{"cdf":"0.999873876571655","Integral":"0.999873876571659"}]}

See Also

BETA - Beta function

BETAI - Incomplete beta function

BETAINV - Inverse of the beta distribution

BETAPDF - Probability density function of the beta distribution