POISSON_DIST

Updated 2023-11-05 12:09:03.517000

Syntax

SELECT [westclintech].[wct].[POISSON_DIST](
  <@X, float,>
 ,<@Mean, float,>
 ,<@Cumulative, bit,>)

Description

Use the scalar function POISSON_DIST to calculate the probability mass function or the lower cumulative probability of a Poisson distribution for a given number of events occurring in a fixed interval given the rate.

Arguments

@Mean

the rate at which events occur during the interval. @Mean must be of a type float or of a type that intrinsically converts to float.

@X

the number of events. @X must be of a type float or of 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.

Return Type

float

Remarks

@X is truncated and the integer value is used.

0 < @X.

0 < @Mean.

Examples

A phone support center averages 43 calls per hour. In this example we calculate the likelihood that it will receive exactly 47 calls in an hour.

SELECT wct.POISSON(   47,     --@X

                      43,     --@Mean

                      'False' --@Cumulative

                  ) as pmf;

This produces the following result.

{"columns":[{"field":"pmf","headerClass":"ag-right-aligned-header","cellClass":"ag-right-aligned-cell"}],"rows":[{"pmf":"0.0484939030267833"}]}

If we wanted to know the likelihood that it would receive up to 47 calls in an hour, we would use the left-tailed cumulative probability.

SELECT wct.POISSON(   47,    --@X

                      43,    --@Mean

                      'True' --@Cumulative

                  ) as cdf;

This produces the following result.

{"columns":[{"field":"cdf","headerClass":"ag-right-aligned-header","cellClass":"ag-right-aligned-cell"}],"rows":[{"cdf":"0.757980664887575"}]}

Since the Poisson distribution is a discrete distribution the cumulative distribution function, in this case, is the sum of the probabilities for each value from 0 to 47. We can use the XLeratorDB SeriesInt table-valued function to generate all the integer values from 0 to 47.

SELECT SUM(wct.POISSON_DIST(SeriesValue, 43, 'False')) as cdf

FROM wct.SeriesInt(0, 47, NULL, NULL, NULL);

This produces the following result.

{"columns":[{"field":"cdf","headerClass":"ag-right-aligned-header","cellClass":"ag-right-aligned-cell"}],"rows":[{"cdf":"0.75798066488758"}]}

The Poisson distribution is a special case of the gamma distribution.

SELECT wct.POISSON_DIST(x, lambda, 'True') as POISSON_DIST,

       wct.GAMMAQ(x + 1, lambda) as GAMMAQ

FROM

(

    VALUES

        (5, 3)

) n (x, lambda);

This produces the following result.

{"columns":[{"field":"POISSON_DIST","headerClass":"ag-right-aligned-header","cellClass":"ag-right-aligned-cell"},{"field":"GAMMAQ","headerClass":"ag-right-aligned-header","cellClass":"ag-right-aligned-cell"}],"rows":[{"POISSON_DIST":"0.916082057968696","GAMMAQ":"0.916082057968696"}]}