symmetric random variable

Let (Ω,,P) be a probability spaceMathworldPlanetmath and X a real random variableMathworldPlanetmath defined on Ω. X is said to be symmetricMathworldPlanetmath if -X has the same distribution functionMathworldPlanetmath as X. A distribution function F:[0,1] is said to be symmetric if it is the distribution function of a symmetric random variable.

Remark. By definition, if a random variable X is symmetric, then E[X] exists (<). Furthermore, E[X]=E[-X]=-E[X], so that E[X]=0. Furthermore, let F be the distribution function of X. If F is continuous at x, then


so that F(x)+F(-x)=1. This also shows that if X has a density function f(x), then f(x)=f(-x).

There are many examples of symmetric random variables, and the most common one being the normal random variables centered at 0. For any random variable X, then the difference ΔX of two independentPlanetmathPlanetmath random variables, identically distributed as X is symmetric.

Title symmetric random variable
Canonical name SymmetricRandomVariable
Date of creation 2013-03-22 16:25:45
Last modified on 2013-03-22 16:25:45
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 8
Author CWoo (3771)
Entry type Definition
Classification msc 60E99
Classification msc 60A99
Defines symmetric distribution function