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| Title of object: |
random variable |
| Canonical Name: |
RandomVariable |
| Type: |
Definition |
| Created on: |
2001-10-25 04:36:21-04 |
| Modified on: |
2002-07-09 16:37:14.304451-04 |
| Classification: |
msc:60-00, msc:62-00 |
Preamble:
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Content:
Let $A$ be a $\sigma$-algebra and $\Omega$ the space of events relative to the experiment. A function $X: (\Omega,A,P()) \to \mathbb{R}$ is a \textbf{random variable} if for every subset $A_r = \{\omega : X(\omega) \leq r\}$, $r \in \mathbb{R}$, the condition $A_r \subset A$ is satisfied.
$X$ is called \textbf{discrete random variable} if the set $ \{X(\omega) : \omega \in \Omega \}$ (i.e. the range of $X$) is finite or countable. Likewise, $X$ is a \textbf{continuous random variable} if the range of $X$ is not countable.
The motivation of a random variable can be explained as follows: consider an experiment such that the outcome is not known beforehand. A random variable can model that experiment and its probabilities of it resulting in different outcomes.
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 random variable.\\
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Note that if $Y = g(X)$, then $Y$ is a new random variable (i.e., any function of a random variable is a random variable). |
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