# random variable

If $(\mathrm{\Omega},\mathcal{A},P)$ is a probability space^{}, then a random variable^{} on $\mathrm{\Omega}$ is a measurable function^{} $X:(\mathrm{\Omega},\mathcal{A})\to S$ to a measurable space^{} $S$ (frequently taken to be the real numbers with the standard measure). The *law* of a random variable is the probability measure $P{X}^{-1}:S\to \mathbb{R}$ defined by $P{X}^{-1}(s)=P({X}^{-1}(s))$.

A random variable $X$ is said to be *discrete* if the set $\{X(\omega ):\omega \in \mathrm{\Omega}\}$ (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 $P(X\in B)=1$ (Note that, as a countable subset of $\mathbb{R}$, $B$ is measurable).

A random variable $Y$ is said to be if it has a cumulative distribution function^{} which is absolutely continuous^{} (http://planetmath.org/AbsolutelyContinuousFunction2).

Example:

Consider the event of throwing a coin. Thus, $\mathrm{\Omega}=\{H,T\}$ 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 |