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In [[statistics|statistical theory]], the [[probability distribution]]s of continuous variables can be expressed in terms of [[probability density function]]s. <ref name="Springer Texts in Statistics">{{Cite journal |last1=Dekking |first1=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]].<ref>{{cite journal |last1=Poyton |first1=A. A. |last2=Varziri |first2=Mohammad Saeed |last3=McAuley |first3=Kimberley B. |last4=MclellanPat James |first4=Pat James |last5=Ramsay |first5=James O. |title=Parameter estimation in continuous-time dynamic models using principal differential analysis |journal=Computers & Chemical Engineering |date=February 15, 2006 |volume=30 |issue=4 |page=698-708 |doi=10.1016/j.compchemeng.2005.11.008}}</ref> 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.
==Discrete variable==
In contrast, a variable is a '''discrete variable''' if and only if there exists a one-to-one correspondence between this variable and <math>\mathbb{N}</math>, the set of [[natural numbers]]. In other words; a discrete variable over a particular interval of real values is one for which, for any value in the range that the variable is permitted to take on, there is a positive minimum distance to the nearest other permissible value. The value of a discrete variable can be obtained by counting, and the number of permitted values is either finite or [[countably infinite]]. Common examples are variables that must be [[Integer|integers]], non-negative integers, positive integers, or only the integers 0 and 1.<ref>{{cite book |last1=van Douwen |first1=Eric |title=Handbook of Set-Theoretic Topology |date=1984 |publisher=Elsevier |___location=North Holland |isbn=978-0-444-86580-9 |pages=113-167}}</ref>
Methods of calculus do not readily lend themselves to problems involving discrete variables. Especially in multivariable calculus, many models rely on the assumption of continuity.<ref>{{cite book |last1=Clogg |first1=Clifford C. |last2=Shockey |first2=James W. |title=Handbook of Multivariate Experimental Psychology |date=1988 |publisher=Springer Publishing Company |___location=Boston, Massachusetts |isbn=978-1-4613-0893-5 |page=337-365}}</ref> Examples of problems involving discrete variables include [[integer programming]].
In statistics, the probability distributions of discrete variables can be expressed in terms of [[probability mass function]]s.<ref name="Springer Texts in Statistics" />
In [[discrete time]] dynamics, the variable ''time'' is treated as discrete, and the equation of evolution of some variable over time is called a [[difference equation]].<ref>{{cite book |last1=Thyagarajan |first1=K.S. |title=Introduction to Digital Signal Processing Using MATLAB with Application to Digital Communications |date=2019 |publisher=Springer Publishing Company |isbn=978-3319760285 |page=21-63 |edition=1}}</ref> For certain discrete-time dynamical systems, the system response can be modeled by solving the difference equation for an analytical solution.
In [[econometrics]] and more generally in [[regression analysis]], sometimes some of the variables being [[empirical]]ly related to each other are 0-1 variables, being permitted to take on only those two values.<ref>{{cite journal |last1=Miller |first1=Jerry L.L. |last2=Erickson |first2=Maynard L. |title=On Dummy Variable Regression Analysis |journal=Sociological Methods & Research |date=May 1974 |volume=2 |issue=4 |page=395-519 |doi=10.1177/004912417400200402}}</ref> The purpose of the discrete values of 0 and 1 is to use the dummy variable as a ‘switch’ that can ‘turn on’ and ‘turn off’ by assigning the two values to different parameters in an equation. A variable of this type is called a [[dummy variable (statistics)|dummy variable]]. If the [[dependent variable]] is a dummy variable, then [[logistic regression]] or [[probit regression]] is commonly employed. In the case of regression analysis, a dummy variable can be used to represent subgroups of the sample in a study (e.g. the value 0 corresponding to a constituent of the control group). <ref>{{cite book |last1=Hardy |first1=Melissa A. |title=Regression with Dummy Variables (Quantitative Applications in the Social Sciences) |date=February 25, 1993 |publisher=Sage Publications, Inc. |___location=Newbury Park |isbn=0803951280 |page=v |edition=1st}}</ref>
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