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Application module: Probability distribution ISO/TS 10303-1274:2004(E)
© ISO

Cover page
Table of contents
Copyright
Foreword
Introduction
1 Scope
2 Normative references
3 Terms, definitions and abbreviations

4 Information requirements
   4.1 Required AM ARMs
   4.2 ARM type definition
   4.3 ARM entity definitions
5 Module interpreted model
   5.1 Mapping specification
   5.2 MIM EXPRESS short listing
     5.2.1 MIM type definition

A MIM short names
B Information object registration
C ARM EXPRESS-G   EXPRESS-G
D MIM EXPRESS-G   EXPRESS-G
E Computer interpretable listings
Bibliography
Index

4 Information requirements

This clause specifies the information requirements for the Probability distribution application module. The information requirements are specified as the Application Reference Model (ARM) of this application module.

NOTE 1  A graphical representation of the information requirements is given in Annex C.

NOTE 2  The mapping specification is specified in 5.1. It shows how the information requirements are met by using common resources and constructs defined or imported in the MIM schema of this application module.

The following EXPRESS specification begins the Probability_distribution_arm schema and identifies the necessary external references.

EXPRESS specification:

*)
SCHEMA Probability_distribution_arm;
(*

4.1 Required AM ARMs

The following EXPRESS interface statements specify the elements imported from the ARMs of other application modules.

EXPRESS specification:

*)
USE FROM Classification_assignment_arm;    --  ISO/TS 10303-1114

USE FROM Measure_representation_arm;    --  ISO/TS 10303-1118

USE FROM Probability_arm;    --  ISO/TS 10303-1252
(*

NOTE 1   The schemas referenced above are specified in the following part of ISO 10303:

Classification_assignment_arm ISO/TS 10303-1114
Measure_representation_arm ISO/TS 10303-1118
Probability_arm ISO/TS 10303-1252

NOTE 2   See Annex C, Figures C.1and C.2 for a graphical representation of this schema.

4.2 ARM type definition

This subclause specifies the ARM type for this application module. The ARM type and definition is specified below.

4.2.1 probability_distribution_classification   EXPRESS-G

The probability_distribution_classification type is an extension of the classification_item type. It adds the data type Probability_distribution to the list of alternate data types.

EXPRESS specification:

*)
TYPE probability_distribution_classification = SELECT BASED_ON classification_item WITH
   (Probability_distribution);
END_TYPE;
(*

4.3 ARM entity definitions

This subclause specifies the ARM entities for this module. Each ARM application entity is an atomic element that embodies a unique application concept and contains attributes specifying the data elements of the entity. The ARM entities and definitions are specified below.

4.3.1 Distribution_by_value   EXPRESS-GMapping table

A Distribution_by_value is a type of Probability_distribution that explicitly lists pairs of random variable values and function values.

Distribution_by_value is used where there is no named distribution which can be used to identify the distribution, for example, when the distribution is derived from observation.

EXPRESS specification:

*)
ENTITY Distribution_by_value
  SUBTYPE OF (Probability_distribution);
  defined_function : Value_function;
  distribution_function : STRING;
END_ENTITY;
(*

Attribute definitions:

defined_function: a list of pairs consisting of a random variable value and the corresponding function value. This represents a numerical approximation to a probability distribution.

NOTE 2   the value of random variable increases strictly monotonically along the list.

EXAMPLE 1   The distribution is the probability generating function for an unfair dice ((1,0.1), (2,0.166),...,(6,0.23)). Here each pair represents the value on the face and the probability of it occuring. The probability of rolling a 1 or a 2 is therefore (0.1 + 0.166).

EXAMPLE 2   The distribution is a cumulative distribution function for the measured length of a component ((2.1, 0.0), (2.2, 0.1), (2.3, 0.6), (2.4, 1.0)) Here each pair represents a length and the probability that the component exceeds that length. The probability that the component is between 2.2m and 2.3m long is therefore 0.5 (0.6 - 0.1).

distribution_function: the type of function which the (random variable, function value) pair list represents.

NOTE 3   Probabilities may be described by a number of different types of functions, such as the cumulative distribution function, the probability density function for continuous distributions or the probability generation function for discrete distributions. Each function is identified through reference data.

4.3.2 Function_value_pair   EXPRESS-GMapping table

A Function_value_pair is a single random variable/value pair, thus defining a single point on the distribution curve of a numerically defined distribution.

EXAMPLE    The distribution function for the tossing of two coins uses the number of heads as the random variable and the probability as value of the function, and has the form of a list of three pairs ((0, 0.25), (1, 0.5), (2, 0.25)) - 0, 1 and 2 being the random variable values, 0.25, 0.5 and 0.25 being the probability values.

EXPRESS specification:

*)
ENTITY Function_value_pair;
  function_value : Probability_function_value;
  variable_value : Random_variable;
END_ENTITY;
(*

Attribute definitions:

function_value: the value of the probability function for the given random variable value.

variable_value: the value of the random variable.

4.3.3 Parameterized_distribution   EXPRESS-GMapping table

A Parameterized_distribution is a type of Probability_distribution that is used to link a named probability distribution to the parameters that define it.

NOTE 1   The parameterization_name of this entity is used to discriminate between alternative parameterizations of the same distribution.

NOTE 2   To calculate a value using a Parameterized_distribution it is necessary to know the general distribution function, the value of the distribution function parameters, and the specific random variable value for which the probability evaluation is required. The Probability Distribution module does not define the formula for the distribution function, and it is assumed that this is defined externally, either through reference information or via a "formula" module. This entity provides the distribution function parameters. The value of the random variable for which the probability is calculated is provided by the Probability_derivation_parameter in the Probability module (see note 2).

EXPRESS specification:

*)
ENTITY Parameterized_distribution
  SUBTYPE OF (Probability_distribution);
  has_parameters : LIST[1:?] OF Probability_distribution_parameter;
  parameterization_name : STRING;
END_ENTITY;
(*

Attribute definitions:

has_parameters: the list of parameters used in the parameterization.

NOTE 3   In many distributions there is a trivial relationship between the mean and variance for the distribution. In such a case it is not necessary to list the parameters.

parameterization_name: the name of the parameterization, distinguishing it from other parameterizations.

4.3.4 Probability_distribution   EXPRESS-GMapping table

A Probability_distribution is a type of Probability_generator that is a probability distribution.

NOTE 1   For a full understanding of probability distribution and the terms used, a textbook on probability theory should be consulted.

NOTE 2   This entity describes a particular probability distribution, rather than the general type of distribution. For example, in coin tossing experiment, the number of heads that may occur is given by a binomial distribution - that is, a type of distribution, and outside the scope of this module. This module provides the description of the distribution of a particular experiment, say, 10 tosses of a particular coin. The actual probability of an outcome, say 6 heads in 10 tosses, is recorded using the probability module.

NOTE 3   The attributes 'name', 'id' and 'description' are inherited from the supertype Representation. The name provides a cue to the particular source of the distribution, such as "A fair coin tossed 10 times", rather than the type of distribution (in this case Binomial) which is given by the Probability_distribution.distribution_name attribute.

EXPRESS specification:

*)
ENTITY Probability_distribution
  ABSTRACT SUPERTYPE
  SUBTYPE OF (Probability_generator);
  distribution_name : OPTIONAL STRING;
  is_continuous : STRING;
  mean : REAL;
  variance : REAL;
END_ENTITY;
(*

Attribute definitions:

distribution_name: the name of the type of distribution. The value of the attribute need not be specified.

NOTE 4   The name is taken from reference data, which is based on the standard mathematical names for distributions, such as "Binomial", "Normal", "Gaussian". Empirically derived distributions will not normally be named.

is_continuous: the text that distinguishes whether the domain of the probability distribution is discrete or continuous.

EXAMPLE 1   The measurement of a person's height is a continuous distribution, since it can take any value between 0 and 3m.

EXAMPLE 2   Coin tossing gives rise to a discrete distribution, since the number of heads cannot be a fraction.

mean: the expectation E(x), which is the "summation" of x.p(x) over all x, where x is the random variable, and p(x) is the probability of its occurrence. The term "summation" covers both literal summation in the case of a discrete distribution, and integration for a continuous distribution.

variance: the real number indicating the square of the standard deviation of the probability distribution.

4.3.5 Probability_distribution_parameter   EXPRESS-GMapping table

A Probability_distribution_parameter is a type of Numerical_item_with_global_unit that is one of the set of values that characterises a probability distribution.

EXAMPLE    The Normal (or Gaussian) distribution has, in the standard parameterisation, two parameters: the mean and the variance

NOTE 1   For many common distributions, the mean and the variance are sufficient to characterize a distribution, and the parameter list may be empty.

Probability_distribution_parameter inherits the 'name' attribute from its supertype, and this is used to identify the name of the parameter within the particular parameterization. The value attribute is also inherited.

EXPRESS specification:

*)
ENTITY Probability_distribution_parameter
  SUBTYPE OF (Numerical_item_with_global_unit);
END_ENTITY;
(*

4.3.6 Probability_function_value   EXPRESS-GMapping table

A Probability_function_value is a type of Numerical_item_with_global_unit that is the value of the probability function at the given random variable value.

NOTE    The value is an inherited attribute. It is not in general a probability value. In some functions for continuous distributions, the probability that the random variable lies between two values is the integral of the function of that range.

EXPRESS specification:

*)
ENTITY Probability_function_value
  SUBTYPE OF (Numerical_item_with_global_unit);
END_ENTITY;
(*

4.3.7 Random_variable   EXPRESS-GMapping table

A Random_variable is a type of Numerical_item_with_global_unit that is the value of a random variable.

NOTE    The value attribute is inherited from the supertype.

EXPRESS specification:

*)
ENTITY Random_variable
  SUBTYPE OF (Numerical_item_with_global_unit);
END_ENTITY;
(*

4.3.8 Value_function   EXPRESS-GMapping table

A Value_function is a numerically defined function that describes the distribution function.

EXPRESS specification:

*)
ENTITY Value_function;
  function_element : LIST[1:?] OF Function_value_pair;
END_ENTITY;
(*

Attribute definitions:

function_element: a single random variable/function value point on the function.



*)
END_SCHEMA;  -- Probability_distribution_arm
(*


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