RANDPOISSON

Updated 2023-10-18 19:31:38.830000

Syntax

SELECT * FROM [westclintech].[wct].[RANDPOISSON](
  <@Rows, int,>
 ,<@lambda, float,>)

Description

Use the table-valued function RANDPOISSON to generate a sequence of random integers from the Poisson distribution for a given @lambda.

Arguments

@lamda

the lambda parameter to the distribution. @lambda 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.

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": "29a7c8ba-ccc0-4da2-a14a-c8ba404c8505", "colName": "Seq", "colDatatype": "int", "colDesc": "A monotonically increasing sequence number"}, {"id": "50b55fe2-bada-48a8-bce5-154846404402", "colName": "X", "colDatatype": "float", "colDesc": "The random variable"}]}

Remarks

@lambda must be greater than zero.

If @lambda is NULL then @lambda 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 from a Poisson distribution with @lambda = 10, COUNT the results, paste them into Excel and graph them.

SELECT X,

       COUNT(*) as [COUNT]

FROM

(

    SELECT X

    FROM wct.RANDPOISSON(   1000000, --@Rows

                            10       --@lambda

                        )

) n

GROUP BY X

ORDER BY 1;

This produces the following result.

In this example we generate 1,000,000 random numbers from a Poisson distribution with @lambda of 4. 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 @lambda as float = 4;

DECLARE @mean as float = @lambda;

DECLARE @var as float = @lambda;

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

DECLARE @skew as float = 1 / @stdev;

DECLARE @kurt as float = 1 / @mean;

SELECT stat,

       [RANDPOISSON],

       [EXPECTED]

FROM

(

    SELECT x.*

    FROM

    (

        SELECT AVG(cast(x as float)) as mean_POISSON,

               STDEVP(x) as stdev_POISSON,

               wct.SKEWNESS_P(x) as skew_POISSON,

               wct.KURTOSIS_P(x) as kurt_POISSON

        FROM wct.RANDPOISSON(@size, @lambda)

    ) n

        CROSS APPLY

    (

        VALUES

            ('RANDPOISSON', 'avg', mean_POISSON),

            ('RANDPOISSON', 'stdev', stdev_POISSON),

            ('RANDPOISSON', 'skew', skew_POISSON),

            ('RANDPOISSON', 'kurt', kurt_POISSON),

            ('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 ([RANDPOISSON], [EXPECTED])

) P;

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

{"columns":[{"field":"stat"},{"field":"RANDPOISSON","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","RANDPOISSON":"3.999409","EXPECTED":"4"},{"stat":"kurt","RANDPOISSON":"0.255689062291338","EXPECTED":"0.25"},{"stat":"skew","RANDPOISSON":"0.501722252577688","EXPECTED":"0.5"},{"stat":"stdev","RANDPOISSON":"1.99860567664535","EXPECTED":"2"}]}