# symmetric random variable

Let $(\mathrm{\Omega},\mathcal{F},P)$ be a probability space^{} and $X$ a real random variable^{} defined on $\mathrm{\Omega}$. $X$ is said to be *symmetric ^{}* if $-X$ has the same distribution function

^{}as $X$. A distribution function $F:\mathbb{R}\to [0,1]$ 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 $x\in \mathbb{R}$, then

$$F(-x)=P(X\le -x)=P(-X\le -x)=P(X\ge x)=1-P(X\le x)=1-F(x),$$ |

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 $\mathrm{\Delta}X$ of two independent^{} 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 |