EXPDIST

Updated 2024-03-07 21:30:53.687000

Syntax

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

Description

Use the scalar function EXPDIST to calculate the probability density function or the upper cumulative distribution function of the Exponential distribution.

The formula for the probability density function is:

f(x;\lambda) = \begin{cases} \lambda  e^{ - \lambda x} & x \ge 0, \\ 0 & x < 0. \end{cases}

The upper cumulative distribution function is:

F(x;\lambda) = \begin{cases} 1-e^{-\lambda x} & x \ge 0, \\ 0 & x < 0. \end{cases}

Arguments

@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.

@X

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

@Lambda

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

Return Type

float

Remarks

@X must be greater than or equal to zero (@X ≥ 0).

@Lambda must be greater than zero (@lambda > 0).

The lower cumulative distribution function is equal to 1 –EXPDIST(@X,@A, 'True').

Examples

Calculate the probability density function:

SELECT wct.EXPDIST(.75, 9.5, 'False');

This produces the following result.

{"columns":[{"field":"column 1","headerClass":"ag-right-aligned-header","cellClass":"ag-right-aligned-cell"}],"rows":[{"column 1":"0.00764496359618383"}]}

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

       wct.EXPDIST(SeriesValue, 1.0, 'False') as [f(x,1.0)],

       wct.EXPDIST(SeriesValue, 1.5, 'False') as [f(x,1.5)],

       wct.EXPDIST(SeriesValue, 2.0, 'False') as [f(x,2.0)],

       wct.EXPDIST(SeriesValue, 2.5, 'False') as [f(x,2.5)]

FROM wct.SeriesFloat(0, 5, .1, NULL, NULL);

This is an EXCEL-generated graph of the results.

Calculate the upper cumulative distribution function:

SELECT wct.EXPDIST(.75, 9.5, 'True');

This produces the following result.

{"columns":[{"field":"column 1","headerClass":"ag-right-aligned-header","cellClass":"ag-right-aligned-cell"}],"rows":[{"column 1":"0.999195266989875"}]}

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

       wct.EXPDIST(SeriesValue, 1.0, 'True') as [f(x,1.0)],

       wct.EXPDIST(SeriesValue, 1.5, 'True') as [f(x,1.5)],

       wct.EXPDIST(SeriesValue, 2.0, 'True') as [f(x,2.0)],

       wct.EXPDIST(SeriesValue, 2.5, 'True') as [f(x,2.5)]

FROM wct.SeriesFloat(0, 5, .1, NULL, NULL);

This is an EXCEL-generated graph of the results.

Calculate the lower cumulative distribution function:

SELECT 1 - wct.EXPDIST(.75, 9.5, 'True');

This produces the following result.

{"columns":[{"field":"column 1","headerClass":"ag-right-aligned-header","cellClass":"ag-right-aligned-cell"}],"rows":[{"column 1":"0.000804733010124625"}]}