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Homerandom variable

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# random variable

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:

## Mathematics Subject Classification

62-00*no label found*60-00

*no label found*11R32

*no label found*03-01

*no label found*20B25

*no label found*

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## Comments

## temporality

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.