Continuous or discrete variable: Difference between revisions

Revision as of 07:03, 3 November 2024

Revision as of 09:39, 7 November 2024

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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 ''[[Measurement|measuring]]'' or ''[[counting]]'', respectively.<ref>{{cite journal |last1=Ali |first1=Zulfiqar |last2=Bhaskar |first2=S. Bala |title=Basic statistical tools in research and data analysis |journal=Indian Journal of Anaesthesia |date=September 2016 |volume=60 |issue=9 |pages=662–669 |doi=10.4103/0019-5049.190623|doi-access=free|pmid=27729694 |pmc=5037948 }}</ref> 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 |pages=82–86 |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.

[[File:Continuous and discrete variables.png|thumb|upright=1.3|right|Variables can be divided into two main categories: [[Categorical variable|qualitative (categorical)]] and quantitative (numerical). Continuous and discrete variables are subcategories of quantitative variables. Note that this schematic is not exhaustive in terms of the types of variables.]]

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==Mixture of continuous and discrete variables==

A mixed multivariate model can contain both discrete and continuous variables. For instance, a simple mixed multivariate model could have a discrete variable <math>x</math>, which only takes on values 0 or 1, and a continuous variable <math>y</math>.<ref>{{cite journal |last1=Olkin |first1=Ingram |last2=Tate |first2=Robert |title=Multivariate Correlation Models with Mixed Discrete and Continuous Variables |journal=The Annals of Mathematical Statistics |date=June 1961 |volume=32 |issue=2 |pages=448–465 |doi=10.1214/aoms/1177705052|doi-access=free }}</ref> An example of a mixed model could be a research study on the risk of psychological disorders based on one binary measure of psychiatric symptoms and one continuous measure of cognitive performance.<ref>{{cite journal |last1=Fitzmaurice |first1=Garrett M. |last2=Laird |first2=Nan M. |title=Regression Models for Mixed Discrete and Continuous Responses with Potentially Missing Values |journal=Biometrics |date=March 1997 |volume=53 |issue=1 |pages=110–122 |doi=10.2307/2533101|jstor=2533101 }}</ref> Mixed models may also involve a single variable that is discrete over some range of the number line and continuous at another range.

In probability theory and statistics, the probability distribution of a mixed random variable consists of both discrete and continuous components. A mixed random variable does not have a [[cumulative distribution function]] that is discrete or everywhere-continuous. An example of a mixed type random variable is the probability of wait time in a queue. The likelihood of a customer experiencing a zero wait time is discrete, while non-zero wait times are evaluated on a continuous time scale.<ref>{{cite journal |last1=Sharma |first1=Shalendra D. |title=On a Continuous/Discrete Time Queueing System with Arrivals in Batches of Variable Size and Correlated Departures |journal=Journal of Applied Probability |date=March 1975 |volume=12 |issue=1 |pages=115–129 |doi=10.2307/3212413|jstor=3212413 }}</ref> In physics (particularly quantum mechanics, where this sort of distribution often arises), [[dirac delta function]]s are often used to treat continuous and discrete components in a unified manner. For example, the previous example might be described by a probability density <math>p(t)=\alpha \delta (t) + g(t)</math>, such that <math>P(t>0)=\int_0^\infty g(t)=1-\alpha</math>, and <math>P(t=0)=\alpha</math>.

==See also==