👁 OpenMath

http://www.openmath.org/cd/s_dist1.ocd

CD Base:

http://www.openmath.org/cd

CD File:

s_dist1.ocd

CD as XML Encoded OpenMath:

s_dist1.omcd

Defines:

mean, moment, sdev, variance

Date:

2004-03-30

Version:

3 (Revision 1)

Review Date:

2006-03-30

Status:

official

 This document is distributed in the hope that it will be useful, 
 but WITHOUT ANY WARRANTY; without even the implied warranty of 
 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.

 The copyright holder grants you permission to redistribute this 
 document freely as a verbatim copy. Furthermore, the copyright
 holder permits you to develop any derived work from this document
 provided that the following conditions are met.
 a) The derived work acknowledges the fact that it is derived from
 this document, and maintains a prominent reference in the 
 work to the original source.
 b) The fact that the derived work is not the original OpenMath 
 document is stated prominently in the derived work. Moreover if
 both this document and the derived work are Content Dictionaries
 then the derived work must include a different CDName element,
 chosen so that it cannot be confused with any works adopted by
 the OpenMath Society. In particular, if there is a Content 
 Dictionary Group whose name is, for example, `math' containing
 Content Dictionaries named `math1', `math2' etc., then you should 
 not name a derived Content Dictionary `mathN' where N is an integer.
 However you are free to name it `private_mathN' or some such. This
 is because the names `mathN' may be used by the OpenMath Society
 for future extensions.
 c) The derived work is distributed under terms that allow the
 compilation of derived works, but keep paragraphs a) and b)
 intact. The simplest way to do this is to distribute the derived
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 If you have questions about this license please contact the OpenMath
 society at http://www.openmath.org.

 Author: OpenMath Consortium
 SourceURL: https://github.com/OpenMath/CDs
 

This CD holds the definitions of the basic statistical functions used on random variables. It is intended to be `compatible' with the MathML elements representing statistical functions.

Role:

application

Description:

This symbol represents a unary function denoting the mean of a distribution. The argument is a univariate function to describe the distribution. That is, if f is the function describing the distribution. The mean is the expression integrate(x*f(x)) w.r.t. x over the range (-infinity,infinity).

Commented Mathematical property (CMP):

mean(f(X)) = int(x*f(x)) w.r.t. x over the range [-infinity,infinity]

Formal Mathematical property (FMP):

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
 <OMA>
 <OMS cd="relation1" name="eq"/>
 <OMA>
 <OMS cd="s_dist1" name="mean"/>
 <OMV name="f"/>
 </OMA>
 <OMA>
 <OMS cd="calculus1" name="defint"/>
 <OMA>
 <OMS cd="interval1" name="interval"/>
	<OMA>
	 <OMS cd="arith1" name="unary_minus"/>
	 <OMS cd="nums1" name="infinity"/>
	</OMA>
	<OMS cd="nums1" name="infinity"/>
 </OMA>
 <OMBIND>
 <OMS cd="fns1" name="lambda"/>
	<OMBVAR>
	 <OMV name="x"/>
	</OMBVAR>
	<OMA>
	 <OMS cd="arith1" name="times"/>
	 <OMV name="x"/>
	 <OMA>
	 <OMV name="f"/>
	 <OMV name="x"/>
	 </OMA>
	</OMA>
 </OMBIND>
 </OMA>
 </OMA>
</OMOBJ>

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
 <apply><csymbol cd="s_dist1">mean</csymbol><ci>f</ci></apply>
 <apply><csymbol cd="calculus1">defint</csymbol>
 <apply><csymbol cd="interval1">interval</csymbol>
 <apply><csymbol cd="arith1">unary_minus</csymbol><csymbol cd="nums1">infinity</csymbol></apply>
 <csymbol cd="nums1">infinity</csymbol>
 </apply>
 <bind><csymbol cd="fns1">lambda</csymbol>
 <bvar><ci>x</ci></bvar>
 <apply><csymbol cd="arith1">times</csymbol>
 <ci>x</ci>
 <apply><ci>f</ci><ci>x</ci></apply>
 </apply>
 </bind>
 </apply>
 </apply>
</math>

eq (mean ( f) , defint (interval (unary_minus (infinity) , infinity) , lambda [ x ] . (times ( x, f ( x) ) ) ) )

s_dist1.mean($f) = calculus1.defint(interval1.interval( -(nums1.infinity), nums1.infinity), fns1.lambda[$x -> $x * $f($x)])

Signatures:

sts

Role:

application

Description:

This symbol represents a unary function denoting the standard deviation of a distribution. The argument is a univariate function to describe the distribution. The standard deviation of a distribution is the arithmetical mean of the squares of the deviation of the distribution from the mean.

Commented Mathematical property (CMP):

The standard deviation of a distribution is the arithmetical mean of the squares of the deviation of the distribution from the mean.

Formal Mathematical property (FMP):

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
 <OMA>
 <OMS cd="relation1" name="eq"/>
 <OMA>
 <OMS cd="s_dist1" name="sdev"/>
 <OMV name="f"/>
 </OMA>
 <OMA>
 <OMS cd="s_dist1" name="mean"/>
 <OMA>
 <OMS cd="arith1" name="power"/>
	<OMA>
	 <OMS cd="arith1" name="minus"/>
	 <OMV name="f"/>
	 <OMA>
	 <OMS cd="s_dist1" name="mean"/>
	 <OMV name="f"/>
	 </OMA>
	</OMA>
	<OMI> 2 </OMI>
 </OMA>
 </OMA>
 </OMA>
</OMOBJ>

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
 <apply><csymbol cd="s_dist1">sdev</csymbol><ci>f</ci></apply>
 <apply><csymbol cd="s_dist1">mean</csymbol>
 <apply><csymbol cd="arith1">power</csymbol>
 <apply><csymbol cd="arith1">minus</csymbol>
 <ci>f</ci>
 <apply><csymbol cd="s_dist1">mean</csymbol><ci>f</ci></apply>
 </apply>
 <cn type="integer">2</cn>
 </apply>
 </apply>
 </apply>
</math>

eq (sdev ( f) , mean (power (minus ( f, mean ( f) ) , 2 ) ) )

s_dist1.sdev($f) = s_dist1.mean(($f - s_dist1.mean($f)) ^ 2)

Signatures:

sts

Role:

application

Description:

This symbol represents a unary function denoting the variance of a distribution. The argument is a function to describe the distribution. That is if f is the function which describes the distribution. The variance of a distribution is the square of the standard deviation of the distribution.

Commented Mathematical property (CMP):

The variance of a distribution is the square of the standard deviation of the distribution.

Formal Mathematical property (FMP):

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
 <OMA>
 <OMS cd="relation1" name="eq"/>
 <OMA>
 <OMS cd="s_dist1" name="variance"/>
 <OMV name="f"/>
 </OMA>
 <OMA>
 <OMS cd="arith1" name="power"/>
 <OMA>
 <OMS cd="s_dist1" name="sdev"/>
	<OMV name="f"/>
 </OMA>
 <OMI> 2 </OMI>
 </OMA>
 </OMA>
</OMOBJ>

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
 <apply><csymbol cd="s_dist1">variance</csymbol><ci>f</ci></apply>
 <apply><csymbol cd="arith1">power</csymbol>
 <apply><csymbol cd="s_dist1">sdev</csymbol><ci>f</ci></apply>
 <cn type="integer">2</cn>
 </apply>
 </apply>
</math>

eq (variance ( f) , power (sdev ( f) , 2 ) )

s_dist1.variance($f) = s_dist1.sdev($f) ^ 2

Signatures:

sts

Role:

application

Description:

This symbol represents a ternary function to denote the i'th moment of a distribution. The first argument should be the degree of the moment (that is, for the i'th moment the first argument should be i), the second argument is the value about which the moment is to be taken and the third argument is a univariate function to describe the distribution. That is, if f is the function which describe the distribution. The i'th moment of f about a is the integral of (x-a)^i*f(x) with respect to x, over the interval (-infinity,infinity).

Commented Mathematical property (CMP):

the i'th moment of f(X) about c = integral of (x-c)^i*f(x) with respect to x, over the interval (-infinity,infinity)

Formal Mathematical property (FMP):

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
 <OMA>
 <OMS cd="relation1" name="eq"/>
 <OMA>
 <OMS cd="s_dist1" name="moment"/>
 <OMV name="i"/>
 <OMV name="c"/>
 <OMV name="f"/>
 </OMA>
 <OMA>
 <OMS cd="calculus1" name="defint"/>
 <OMA>
 <OMS cd="interval1" name="interval"/>
	<OMA>
	 <OMS cd="arith1" name="unary_minus"/>
	 <OMS cd="nums1" name="infinity"/>
	</OMA>
	<OMS cd="nums1" name="infinity"/>
 </OMA>
 <OMBIND>
 <OMS cd="fns1" name="lambda"/>
	<OMBVAR>
	 <OMV name="x"/>
	</OMBVAR>
 <OMA>
 <OMS cd="arith1" name="times"/>
	 <OMA>
	 <OMS cd="arith1" name="power"/>
	 <OMA>
	 <OMS cd="arith1" name="minus"/>
	 <OMV name="x"/>
	 <OMV name="c"/>
	 </OMA>
	 <OMV name="i"/>
	 </OMA>
	 <OMA>
	 <OMV name="f"/>
	 <OMV name="x"/>
	 </OMA>
 </OMA>
 </OMBIND>
 </OMA>
 </OMA>
</OMOBJ>

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
 <apply><csymbol cd="s_dist1">moment</csymbol><ci>i</ci><ci>c</ci><ci>f</ci></apply>
 <apply><csymbol cd="calculus1">defint</csymbol>
 <apply><csymbol cd="interval1">interval</csymbol>
 <apply><csymbol cd="arith1">unary_minus</csymbol><csymbol cd="nums1">infinity</csymbol></apply>
 <csymbol cd="nums1">infinity</csymbol>
 </apply>
 <bind><csymbol cd="fns1">lambda</csymbol>
 <bvar><ci>x</ci></bvar>
 <apply><csymbol cd="arith1">times</csymbol>
 <apply><csymbol cd="arith1">power</csymbol>
 <apply><csymbol cd="arith1">minus</csymbol><ci>x</ci><ci>c</ci></apply>
 <ci>i</ci>
 </apply>
 <apply><ci>f</ci><ci>x</ci></apply>
 </apply>
 </bind>
 </apply>
 </apply>
</math>

eq (moment ( i, c, f) , defint (interval (unary_minus (infinity) , infinity) , lambda [ x ] . (times (power (minus ( x, c) , i) , f ( x) ) ) ) )

s_dist1.moment($i, $c, $f) = calculus1.defint(interval1.interval( -(nums1.infinity), nums1.infinity), fns1.lambda[$x -> ($x - $c) ^ $i * $f($x)])

Signatures:

sts