LAPLACE

Updated 2024-03-08 20:37:49.457000

Syntax

SELECT [westclintech].[wct].[LAPLACE] (
  <@X, float,>
 ,<@A, float,>
 ,<@B, float,>
 ,<@Cumulative, bit,>)

Description

Use the scalar function LAPLACE to calculate the probability density function or the lower cumulative distribution function of the Laplace distribution.

The formula for the probability density function is:

f(x \mid \mu, b) = \frac{1}{2b} \exp\left( -\frac{|x - \mu|}{b} \right)

The lower cumulative distribution function is:

F(x)=\frac{1}{2}+\frac{1}{2}\text{ sgn}(x-\mu)\left(1-\text{exp}\left(-\frac{|x - \mu|}{b}\right)\right)

Arguments

@A

is the location parameter. @A is an expression of type float or of a type that implicitly converts to float.

@B

is the scale parameter. @B is an expression of type float or of a type that implicitly converts to float.

@X

is the value to be evaluated. @X is an expression of type float or of a type that implicitly converts to float.

@Cumulative

is a logical value that determines if the probability density function ('False', 0) or the cumulative distribution function ('True', 1) is being calculated. @Cumulative is an expression of type bit or of a type that implicitly converts to bit.

Return Type

float

Remarks

@B must be greater than zero (@B > 0).

The upper cumulative distribution function is equal to 1 –LAPLACE(@X, @A, @B, 'True').

Examples

Calculate the probability density function:

SELECT wct.LAPLACE(0.5, 0, 1, 'False');

This produces the following result.

{"columns":[{"field":"column 1","headerClass":"ag-right-aligned-header","cellClass":"ag-right-aligned-cell"}],"rows":[{"column 1":"0.303265329856317"}]}

You can use the SeriesFloat function from the XLeratorDB/math library to generate a dataset which can be pasted into EXCEL to generate a graph of the probability density function.

SELECT SeriesValue,

       wct.LAPLACE(SeriesValue, 0, 1, 'False') as [f(x,0,0.5)],

       wct.LAPLACE(SeriesValue, 0, 2, 'False') as [f(x,0,1)],

       wct.LAPLACE(SeriesValue, 0, 4, 'False') as [f(x,0,2)],

       wct.LAPLACE(SeriesValue, -5, 4, 'False') as [f(x,-5,1)]

FROM wct.SeriesFloat(-10, 10, .1, NULL, NULL);

This is an EXCEL-generated graph of the results.

Calculate the lower cumulative distribution function:

SELECT wct.LAPLACE(0.5, 0, 1, 'True');

This produces the following result.

{"columns":[{"field":"column 1","headerClass":"ag-right-aligned-header","cellClass":"ag-right-aligned-cell"}],"rows":[{"column 1":"0.696734670143683"}]}

You can use the SeriesFloat function from the XLeratorDB/math library to generate a dataset which can be pasted into EXCEL to generate a graph of the cumulative distribution function.

SELECT SeriesValue,

       wct.LAPLACE(SeriesValue, 0, 1, 'True') as [f(x,0,0.5)],

       wct.LAPLACE(SeriesValue, 0, 2, 'True') as [f(x,0,1)],

       wct.LAPLACE(SeriesValue, 0, 4, 'True') as [f(x,0,2)],

       wct.LAPLACE(SeriesValue, -5, 4, 'True') as [f(x,-5,1)]

FROM wct.SeriesFloat(-10, 10, .1, NULL, NULL);

This is an EXCEL-generated graph of the results.

Calculate the upper cumulative distribution function:

SELECT 1 - wct.LAPLACE(0.5, 0, 1, 'True');

This produces the following result.

{"columns":[{"field":"column 1","headerClass":"ag-right-aligned-header","cellClass":"ag-right-aligned-cell"}],"rows":[{"column 1":"0.303265329856317"}]}