random variable

If (Ω,𝒜,P) is a probability spaceMathworldPlanetmath, then a random variableMathworldPlanetmath on Ω is a measurable functionMathworldPlanetmath X:(Ω,𝒜)S to a measurable spaceMathworldPlanetmathPlanetmath S (frequently taken to be the real numbers with the standard measure). The law of a random variable is the probability measure PX-1:S defined by PX-1(s)=P(X-1(s)).

A random variable X is said to be discrete if the set {X(ω):ωΩ} (i.e. the range of X) is finite or countable. A more general version of this definition is as follows: A random variable X is discrete if there is a countable subset B of the range of X such that P(XB)=1 (Note that, as a countable subset of , B is measurable).

A random variable Y is said to be if it has a cumulative distribution functionMathworldPlanetmathPlanetmathPlanetmath which is absolutely continuousMathworldPlanetmath (http://planetmath.org/AbsolutelyContinuousFunction2).


Consider the event of throwing a coin. Thus, Ω={H,T} where H is the event in which the coin falls head and T the event in which falls tails. Let X=number of tails in the experiment. Then X is a (discrete) random variable.

Title random variable
Canonical name RandomVariable
Date of creation 2013-03-22 11:53:10
Last modified on 2013-03-22 11:53:10
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 21
Author mathcam (2727)
Entry type Definition
Classification msc 62-00
Classification msc 60-00
Classification msc 11R32
Classification msc 03-01
Classification msc 20B25
Related topic DistributionFunction
Related topic DensityFunction
Related topic GeometricDistribution2
Defines discrete random variable
Defines continuous random variable
Defines law of a random variable