# random variable

If $(\Omega,\mathcal{A},P)$ is a probability space, then a on $\Omega$ is a measurable function $X:(\Omega,\mathcal{A})\to S$ 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 $PX^{-1}:S\to\mathbb{R}$ defined by $PX^{-1}(s)=P(X^{-1}(s))$.

A random variable $X$ is said to be discrete if the set $\{X(\omega):\omega\in\Omega\}$ (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, $\Omega=\{H,T\}$ 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