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XLeratorDB/math Documentation

SQL Server random exponential distribution


RANDEXP

Updated: 31 March 2014


Use the table-valued function RANDEXP to generate a sequence of random numbers from an exponential distribution with rate @lambda.
Syntax
SELECT * FROM [wctMath].[wct].[RANDEXP](
  <@Rows, int,>
 ,<@lambda, float,>)
Arguments
@Rows
the number of rows to generate. @Rows must be of the type int or of a type that implicitly converts to int.
@lambda
the rate parameter. @lambda must be of the type float or of a type that implicitly converts to float.
Return Types
RETURNS TABLE (
      [Seq] [int] NULL,
      [X] [float] NULL
)
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 rounded to one decimal place from an exponential distribution with @lambda = 1, COUNT each result, paste then into Excel, and graph them.
SELECT
   X,
   COUNT(*) as [COUNT]
FROM (
   SELECT
      ROUND(X,1) as X
   FROM
      wct.RANDEXP(
         1000000, --@Rows
         1        --@lambda
      )
   )n
GROUP BY
   X
ORDER BY
   X

This produces the following result.

In this example we generate 1,000,000 random numbers from an exponential distribution with @lambda of 2. 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 @mu as float = 2
DECLARE @mean as float = 1e+00/@mu
DECLARE @var as float = POWER(@mu,-2)
DECLARE @stdev as float = SQRT(@var)
DECLARE @skew as float = 2
DECLARE @kurt as float = 6
 
SELECT
   stat,
   [RANDEXP],
   [EXPECTED]
FROM (
   SELECT
      x.*
   FROM (
      SELECT
         AVG(x) as mean_EXP,
         STDEVP(x) as stdev_EXP,
         wct.SKEWNESS_P(x) as skew_EXP,
         wct.KURTOSIS_P(x) as kurt_EXP
      FROM
         wct.RANDEXP(@size,@mu)
      )n
   CROSS APPLY(
      VALUES
         ('RANDEXP','avg', mean_EXP),
         ('RANDEXP','stdev', stdev_EXP),
         ('RANDEXP','skew', skew_EXP),
         ('RANDEXP','kurt', kurt_EXP),
         ('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([RANDEXP],[EXPECTED])) P

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

stat
RANDEXP
EXPECTED
avg
0.500344868
0.5
kurt
5.918523715
6
skew
1.995981452
2
stdev
0.500702317
0.5

 

See Also

 



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