random variable

If $(\mathrm{\Omega },𝒜,P)$ is a probability space, then a random variable on $\mathrm{\Omega }$ is a measurable function $X:(\mathrm{\Omega },𝒜)\to S$ 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 $P{X}^{-1}:S\to ℝ$ 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 $ℝ$, $B$ 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