Continuous and discrete variables

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In mathematics, variables are either continuous or discrete, depending on whether or not there are gaps between a value that the variable could take on and any other permitted values. A variable is continuous if there are no such gaps, so the variable can range continuously over the potential values; otherwise the variable is discrete.

Continuous variables[edit]

A continuous variable is one whose value must be a member of a set such that, if the values a and b are members of the set, then every number between a and b is also in the set. A common example is a variable that is defined over some interval of the real number line. The number of permitted values is uncountable.

Methods of calculus are often used in problems in which the variables are continuous, for example in continuous optimization problems.

In statistical theory, the probability distributions of continuous variables can be expressed in terms of probability density functions.

In continuous-time dynamics, the variable time is treated as continuous, and the equation describing the evolution of some variable over time is a differential equation.

Discrete variables[edit]

In contrast, a discrete variable is one for which, for any two values that the variable is permitted to take on, not all values between them are permitted. The number of permitted values is either finite or countably infinite. Common examples are variables that must be integers, non-negative integers, positive integers, or only the integers 0 and 1.

Methods of calculus do not readily lend themselves to problems involving discrete variables. 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 functions.

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.

In econometrics and more generally in regression analysis, sometimes some of the variables being empirically related to each other are 0-1 variables, being permitted to take on only those two values. A variable of this type is called a dummy variable. If the dependent variable is a dummy variable, then logistic regression or probit regression is commonly employed.

See also[edit]