Continuous or discrete variable: Difference between revisions

Revision as of 02:58, 3 December 2023

Revision as of 02:30, 4 December 2023

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{{Probability fundamentals}}

In [[mathematics]] and [[statistics]], a quantitative [[variable (mathematics)|variable]] may be '''continuous''' or '''discrete''' if they are typically obtained by ''measuring'' or ''[[counting]]'', respectively. If it can take on two particular [[real number|real]] values such that it can also take on all real values between them (including values that are arbitrarily or infinitesimally close together), the variable is continuous in that [[Interval (mathematics)|interval]].<ref>{{cite journal |last1=Kaliyadan |first1=Feroze |last2=Kulkarni |first2=Vinay |title=Types of Variables, Descriptive Statistics, and Sample Size |journal=Indian Dermatology Online Journal |date=January 2019 |volume=10 |issue=1 |pagepages=82-8682–86 |doi=10.4103/idoj.IDOJ_468_18 |pmid=30775310 |pmc=6362742 |doi-access=free }}</ref> If it can take on a value such that there is a non-[[infinitesimal]] gap on each side of it containing no values that the variable can take on, then it is discrete around that value.<ref>K.D. Joshi, ''Foundations of Discrete Mathematics'', 1989, New Age International Limited, [https://books.google.com/books?id=RM1D3mFw2u0C&dq=continuous+discrete+variable+math&pg=PA7], page 7.</ref> In some contexts a variable can be discrete in some ranges of the [[number line]] and continuous in others.

==Continuous variable==

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Methods of [[calculus]] are often used in problems in which the variables are continuous, for example in continuous [[optimization]] problems.<ref>{{Cite book |last1=Griva |first1=Igor |url=https://www.worldcat.org/oclc/236082842 |title=Linear and nonlinear optimization |last2=Nash |first2=Stephen |last3=Sofer |first3=Ariela|author3-link= Ariela Sofer |publisher=Society for Industrial and Applied Mathematics |year=2009 |isbn=978-0-89871-661-0 |edition=2nd |___location=Philadelphia |pages=7 |language=en |oclc=236082842}}</ref>

In [[statistics|statistical theory]], the [[probability distribution]]s of continuous variables can be expressed in terms of [[probability density function]]s. <ref>{{Cite journal |lastlast1=Dekking |firstfirst1=Frederik Michel |last2=Kraaikamp |first2=Cornelis |last3=Lopuhaä |first3=Hendrik Paul |last4=Meester |first4=Ludolf Erwin |date=2005 |title=A Modern Introduction to Probability and Statistics |url=https://doi.org/10.1007/1-84628-168-7 |journal=Springer Texts in Statistics |language=en |doi=10.1007/1-84628-168-7 |isbn=978-1-85233-896-1 |issn=1431-875X}}</ref>

In [[continuous time|continuous-time]] [[dynamical system|dynamics]], the variable ''time'' is treated as continuous, and the equation describing the evolution of some variable over time is a [[differential equation]]. The [[instantaneous rate of change]] is a well-defined concept that takes the ratio of the change in the dependent variable to the independent variable at a specific instant.