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random variable (Definition)

If $ (\Omega,\mathcal {A},P)$ is a probability space, then a random variable 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 continuous if it has a cumulative distribution function which is absolutely continuous.

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.



"random variable" is owned by mathcam. [ full author list (2) | owner history (1) ]
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See Also: distribution function, density function, geometric distribution

Also defines:  discrete random variable, continuous random variable, law of a random variable
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Cross-references: event, cumulative distribution function, measurable, subset, countable, finite, range, discrete, probability measure, measure, real numbers, measurable space, measurable function, probability space
There are 138 references to this entry.

This is version 16 of random variable, born on 2001-10-25, modified 2007-02-25.
Object id is 485, canonical name is RandomVariable.
Accessed 30160 times total.

Classification:
AMS MSC: 60-00 (Probability theory and stochastic processes :: General reference works )
 62-00 (Statistics :: General reference works )
Pending Errata and Addenda
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temporality by rabeldin on 2001-11-11 07:04:27
The use of "beforehand" suggests that the appropriate model is a Cartesian product of the sample space and a continuum representing time. There is an implicit reference to the epoch of observation which may be missed by some readers.
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