Title: distribution function
Long Title: IUPAC Gold Book - distribution function
DOI: 10.1351/goldbook.D01815
Status: current

Definition
A normalized function giving the relative amount of a portion of a polymeric substance with a specific value, or a range of values, of a random variable or variables.

Notes
1) Distribution functions may be discrete, i.e. take on only certain specified values of the random variable(s), or continuous, i.e. take on any intermediate value of the random variable(s), in a given range. Most distributions in polymer science are intrinsically discrete, but it is often convenient to regard them as continuous or to use distribution functions that are inherently continuous.
2) Distribution functions may be integral (or cumulative), i.e. give the proportion of the population for which a random variable is less than or equal to a given value. Alternatively they may be differential distribution functions (or probability density functions), i.e. give the (maybe infinitesimal) proportion of the population for which the random variable(s) is (are) within a (maybe infinitesimal) interval of its (their) range(s).
3) Normalization requires that: (i) for a discrete differential distribution function, the sum of the function values over all possible values of the random variable(s) be unity; (ii) for a continuous differential distribution function, the integral over the entire range of the random variable(s) be unity; (iii) for an integral (cumulative) distribution function, the function value at the upper limit of the random variable(s) be unity.

Related Terms
- Normalization: https://goldbook.iupac.org//terms/view/NT07086
- polymeric: https://goldbook.iupac.org//terms/view/M03667
- probability density: https://goldbook.iupac.org//terms/view/P04856
- variable: https://goldbook.iupac.org//terms/view/V06600

Source
- Purple Book, 1st ed., p. 55 (http://old.iupac.org/publications/books/author/metanomski.html)

Other Outputs
- html: https://goldbook.iupac.org/terms/view/D01815/html
- json: https://goldbook.iupac.org/terms/view/D01815/json
- xml: https://goldbook.iupac.org/terms/view/D01815/xml

Citation: Citation: 'distribution function' in IUPAC Compendium of Chemical Terminology, 5th ed. International Union of Pure and Applied Chemistry; 2025. Online version 5.0.0, 2025. 10.1351/goldbook.D01815
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Accessed: 2026-04-08T02:04:25+00:00