OpenMath Content Dictionary: s_dist1

Canonical URL:
http://www.openmath.org/cd/s_dist1.ocd
CD File:
s_dist1.ocd
CD as XML Encoded OpenMath:
s_dist1.omcd
Defines:
mean, moment, sdev, variance
Date:
2001-03-12
Version:
2
Review Date:
2003-04-01
Status:
official
Uses CD:
relation1, calculus1, interval1, arith1, nums1, fns1, arith1, fns1

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.


mean

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>
  <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>

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

Signatures:
sts


[Next: sdev] [Last: moment] [Top]

sdev

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>
  <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>

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

Signatures:
sts


[Next: variance] [Previous: mean] [Top]

variance

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>
  <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>

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

Signatures:
sts


[Next: moment] [Previous: sdev] [Top]

moment

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>
  <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>

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

Signatures:
sts


[First: mean] [Previous: variance] [Top]