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Continuous random variable

by , PhD

A continuous random variable is a random variable having two main characteristics: 1) the set of values it can take is not countable; 2) its cumulative distribution function can be obtained by integrating a function called probability density function.

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Continuous random variables are sometimes also called absolutely continuous.


The following is a formal definition.

Definition A random variable X is said to be continuous if and only if the probability that its realization will belong to an interval $left[ a,b
ight] $ can be expressed as an integral:[eq1]where the integrand function [eq2] is called the probability density function of X.

Note that, as a consequence of this definition, the cumulative distribution function of $X~$is[eq3]which explains the introductory definition we have given.


Let us give some examples (go to this lecture if you need to revise the basics of integration).

Example 1

Let X be a continuous random variable that can take any value in the interval $left[ 0,1
ight] $. Let its probability density function be[eq4]

Then, for example, the probability that X takes a value between $1/2$ and 1 can be computed as follows:[eq5]

Example 2

Let X be a continuous random variable that can take any value in the interval $left[ 0,3
ight] $ with probability density function[eq6]

The probability that the realization of X will belong to the interval $left[ 0,1
ight] $ is[eq7]


Continuous random variables have some interesting properties:

Common continuous distributions

The next table contains some examples of continuous distributions that are frequently encountered in probability theory and statistics.

Name of the continuous distribution Support
Uniform All the real numbers in the interval [0,1]
Normal The whole set of real numbers
Chi-square The set of all non-negative real numbers

More details

Continuous random variables are discussed in more detail in the lecture entitled Random variables.

You can also read a brief introduction to the probability density function, including some examples, in the glossary entry entitled Probability density function.

Keep reading the glossary

Next entry: Absolutely continuous random vector

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