RANDFDIST

Updated 2023-10-18 15:50:18.283000

Syntax

SELECT * FROM [westclintech].[wct].[RANDFDIST](
  <@Rows, int,>
 ,<@df1, float,>
 ,<@df2, float,>)

Description

Use the table-valued function RANDFDIST to generate a sequence of random numbers from an F-distribution with the degree of freedom parameters @df1 and @df2.

Arguments

@df1

the first degree of freedom parameter. @df1 must be of the type float or of a type that implicitly converts to float.

@Rows

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

@df2

the second degree of freedom parameter. @df2 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": "239b9d4d-f22a-4521-9158-c1ddda96a996", "colName": "Seq", "colDatatype": "int", "colDesc": "A monotonically increasing sequence number"}, {"id": "9a495c1f-b3ed-49c0-9b82-0efca06f904d", "colName": "X", "colDatatype": "float", "colDesc": "The random variable"}]}

Remarks

@df1 must be greater than zero.

@df2 must be greater than zero.

If @df1 is NULL then @df1 is set to 1.

If @df2 is NULL then @df2 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 an F-distribution with @df1 = 100 and @df2 = 100, COUNT the results, paste then into Excel and graph them.

SELECT X,

       COUNT(*) as [COUNT]

FROM

(

    SELECT ROUND(X, 1) as X

    FROM wct.RANDFDIST(   1000000, --@Rows

                          100,     --@df1

                          100      --@df2

                      )

) n

GROUP BY X

ORDER BY X;

This produces the following result.

In this example we generate 1,000,000 random numbers from an F-distribution with @df1 of 100 and a @df2 of 100. 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 @d1 as float = 100;
DECLARE @d2 as float = 100;
DECLARE @mean as float =
CASE
WHEN @d2 <= 2 THEN NULL
ELSE @d2/(@d2-2)
END;
DECLARE @var as float =
CASE
WHEN @d2 <=4 THEN NULL
ELSE (2*POWER(@d2,2)*(@d1+@d2-2))/(@d1*POWER(@d2-2,2)*(@d2-4)) END;
DECLARE @stdev as float =
CASE
WHEN @d2 <=4 THEN NULL
ELSE SQRT(@var) END;
DECLARE @skew as float =
CASE
WHEN @d2 <= 6 THEN NULL
ELSE ((2*@d1+@d2-2)*SQRT(8*(@d2-4)))/((@d2-6)*SQRT(@d1*(@d1+@d2-2)))
END;
DECLARE @kurt as float =
CASE
WHEN @d2 <= 8 THEN NULL
ELSE 12*((@d1*(5*@d2-22)*(@d1+@d2-2)+(@d2-4)*POWER(@d2-2,2))/(@d1*(@d2-6)*(@d2-8)*(@d1+@d2-2)))
END;
 
SELECT
   stat,
   [RANDFDIST],
   [EXPECTED]
FROM (
   SELECT
      x.*
   FROM (
      SELECT
         AVG(x) as mean_FDIST,
         STDEVP(x) as stdev_FDIST,
         wct.SKEWNESS_P(x) as skew_FDIST,
         wct.KURTOSIS_P(x) as kurt_FDIST
      FROM
         wct.RANDFDIST(@size,@d1,@d2)
      )n
   CROSS APPLY(
      VALUES
         ('RANDFDIST','avg', mean_FDIST),
         ('RANDFDIST','stdev', stdev_FDIST),
         ('RANDFDIST','skew', skew_FDIST),
         ('RANDFDIST','kurt', kurt_FDIST),
         ('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([RANDFDIST],[EXPECTED])) P;

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

{"columns":[{"field":"stat"},{"field":"RANDFDIST","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","RANDFDIST":"1.02031428169584","EXPECTED":"1.02040816326531"},{"stat":"kurt","RANDFDIST":"0.742223766504818","EXPECTED":"0.727888318896645"},{"stat":"skew","RANDFDIST":"0.625057408675161","EXPECTED":"0.624361945733466"},{"stat":"stdev","RANDFDIST":"0.207189184603578","EXPECTED":"0.207245877669285"}]}