RANDBETA

Updated 2023-10-18 15:29:15.583000

Syntax

SELECT * FROM [westclintech].[wct].[RANDBETA](
  <@Rows, int,>
 ,<@a, float,>
 ,<@b, float,>)

Description

Use the table-valued function RANDBETA to generate q sequence of random numbers from the beta distribution with two positive shape parameters @a and @b.

Arguments

@Rows

the number of rows to generate. @Rows must be of the type int or of a type that implicitly converts to int.

@b

the second shape parameter. @b must be of the type float or of a type that implicitly converts to float.

@a

the first shape parameter. @a must be of the type float or of a type that implicitly converts to float.

Return Type

table

{"columns": [{"field": "colName", "headerName": "Name", "header": "name"}, {"field": "colDatatype", "headerName": "Type", "header": "type"}, {"field": "colDesc", "headerName": "Description", "header": "description", "minWidth": 1000}], "rows": [{"id": "ea943357-396f-44ff-a453-f00a51e3f14e", "colName": "Seq", "colDatatype": "int", "colDesc": "A monotonically increasing sequence number"}, {"id": "2bc66060-0ce2-4ae7-b54f-916d3a5d292e", "colName": "X", "colDatatype": "float", "colDesc": "The random variable"}]}

Remarks

@a must be greater than zero.

@b must be greater than zero.

If @a is NULL then @a is set to 1.

If @b is NULL then @b is set to 1.

If @Rows is less than 1 then no rows are returned.

Examples

In this example we create a sequence 1,000,000 random numbers rounded to two decimal places from a beta distribution with @a = 0.5 and @b = 0.5, COUNT the result, paste them into Excel and graph them.

SELECT X,

       COUNT(*) as [COUNT]

FROM

(

    SELECT ROUND(x, 2) as X

    FROM wct.RANDBETA(   1000000, --@Rows

                         0.5,     --@a

                         0.5      --@b

                     )

) n

GROUP BY X

ORDER BY 1;

This produces the following result.

In this example we generate 1,000,000 random numbers from a beta distribution with @a of 2 and @b of 5. We calculate the mean, standard deviation, skewness, and excess kurtosis from the resultant table and compare those values to the expected values for the distribution.

DECLARE @size as int = 1000000;

DECLARE @a as float = 2;

DECLARE @b as float = 5;

DECLARE @mean as float = @a / (@a + @b);

DECLARE @var as float = (@a * @b) / (POWER(@a + @b, 2) * (@a + @b + 1));

DECLARE @stdev as float = SQRT(@var);

DECLARE @skew as float = (2 * (@b - @a) * SQRT(@a + @b + 1)) / ((@a + @b + 2) * 

          SQRT(@a * @b));

DECLARE @kurt as float

    = (6 * (POWER(@a - @b, 2) * (@a + @b + 1) - @a * @b * (@a + @b + 2))) / (@a * 

              @b * (@a + @b + 2) * (@a + @b + 3));

SELECT stat,

       [RANDBETA],

       [EXPECTED]

FROM

(

    SELECT x.*

    FROM

    (

        SELECT AVG(x) as mean_BETA,

               STDEVP(x) as stdev_BETA,

               wct.SKEWNESS_P(x) as skew_BETA,

               wct.KURTOSIS_P(x) as kurt_BETA

        FROM wct.RANDBETA(@size, @a, @b)

    ) n

        CROSS APPLY

    (

        VALUES

            ('RANDBETA', 'avg', mean_BETA),

            ('RANDBETA', 'stdev', stdev_BETA),

            ('RANDBETA', 'skew', skew_BETA),

            ('RANDBETA', 'kurt', kurt_BETA),

            ('EXPECTED', 'avg', @mean),

            ('EXPECTED', 'stdev', @stdev),

            ('EXPECTED', 'skew', @skew),

            ('EXPECTED', 'kurt', @kurt)

    ) x (fn_name, stat, val_stat)

) d

PIVOT

(

    sum(val_stat)

    FOR fn_name in ([RANDBETA], [EXPECTED])

) P;

This produces the following result (your result will be different).

{"columns":[{"field":"stat"},{"field":"RANDBETA","headerClass":"ag-right-aligned-header","cellClass":"ag-right-aligned-cell"},{"field":"EXPECTED","headerClass":"ag-right-aligned-header","cellClass":"ag-right-aligned-cell"}],"rows":[{"stat":"avg","RANDBETA":"0.28577495782875","EXPECTED":"0.285714285714286"},{"stat":"kurt","RANDBETA":"-0.119163723672499","EXPECTED":"-0.12"},{"stat":"skew","RANDBETA":"0.597784901594543","EXPECTED":"0.596284793999944"},{"stat":"stdev","RANDBETA":"0.159684726349007","EXPECTED":"0.159719141249985"}]}