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If $(\Omega,\mc{A},P)$ is a probability space, then a \textbf{random variable} on $\Omega$ is a measurable function $X: (\Omega,\mc{A}) \to S$ to a measurable space $S$ (frequently taken to be the real numbers with the standard measure). The \emph{law} of a random variable is the probability measure $PX^{-1}:S\to \R$ defined by $PX^{-1}(s)=P(X^{-1}(s))$.
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If $(\Omega,\mc{A},P)$ is a probability space, then a \textbf{random variable} on $\Omega$ is a measurable function $X: (\Omega,\mc{A}) \to S$ to a measurable space $S$ (frequently taken to be the real numbers with the standard measure). The \emph{law} of a random variable is the probability measure $PX^{-1}:\Omega\to \R$ defined by $PX^{-1}(s)=P(X^{-1}(s))$.
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| A random variable $X$ is said to be \emph{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 $X$ is said to be \emph{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). |
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| A random variable $Y$ is said to be \emph{\PMlinkescapetext{continuous}} if it has a cumulative distribution function which is \PMlinkname{absolutely continuous}{AbsolutelyContinuousFunction2}. |
A random variable $Y$ is said to be \emph{\PMlinkescapetext{continuous}} if it has a cumulative distribution function which is \PMlinkname{absolutely continuous}{AbsolutelyContinuousFunction2}. |
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| Example: |
Example: |
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| 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. |
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. |
Let $X=$number of tails in the experiment. Then $X$ is a (discrete) random variable. |
