CAUCHY

Updated 2024-03-07 20:14:11.783000

Syntax

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

Description

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

The formula for the probability density function is:

f(x; x_0,\gamma) = \frac{1}{\pi\gamma \left[1 + \left(\frac{x - x_0}{\gamma}\right)^2\right]} = { 1 \over \pi } \left[ { \gamma \over (x - x_0)^2 + \gamma^2  } \right]

The lower cumulative distribution function is:

F(x; x_0,\gamma)=\frac{1}{\pi} \arctan\left(\frac{x-x_0}{\gamma}\right)+\frac{1}{2}

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 shape 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 –CAUCHY(@X,@A,@B,'True').

Examples

Calculate the probability density function:

SELECT wct.CAUCHY(1, 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":"0.159154943091895"}]}

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.CAUCHY(SeriesValue, 0, 0.5, 'False') as [f(x,0,0.5)],

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

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

       wct.CAUCHY(SeriesValue, -2, 1, 'False') as [f(x,-2,1)],

       wct.CAUCHY(SeriesValue, -2, 2, 'False') as [f(x,-2,2)]

FROM wct.SeriesFloat(-5, 5, .1, NULL, NULL);

This is an EXCEL-generated graph of the results.

Calculate the lower cumulative distribution function:

SELECT wct.CAUCHY(1, 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":".75"}]}

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.CAUCHY(SeriesValue, 0, 0.5, 'True') as [f(x,0,0.5)],

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

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

       wct.CAUCHY(SeriesValue, -2, 1, 'True') as [f(x,-2,1)],

       wct.CAUCHY(SeriesValue, -2, 2, 'True') as [f(x,-2,2)]

FROM wct.SeriesFloat(-5, 5, .1, NULL, NULL);

This is an EXCEL-generated graph of the results

Calculate the upper cumulative distribution function:

SELECT 1 - wct.CAUCHY(1, 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":".25"}]}