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'conditional probability'
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| Title of object: |
conditional probability |
| Canonical Name: |
ConditionalProbability |
| Type: |
Definition |
| Created on: |
2002-02-18 02:56:03 |
| Modified on: |
2002-02-18 02:56:03 |
| Classification: |
msc:03B48 |
Revision comment (for changes between this and next version):
| changes for correction #7166 |
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Content:
Let $(\Omega, \borel, \mu)$ be a probability space, and let $X$ and $Y$ be random variables on $\Omega$ with joint probability distribution $\mu(X,Y) \defined \mu(X \cap Y)$.
The \emph{conditional probability} of $X$ given $Y$ is defined as
\begin{equation}
\mu(X|Y) \defined \frac{\mu(X \cap Y)}{\mu(Y)}.
\end{equation}
In general,
\begin{equation}
\mu(X|Y)\mu(Y) = \mu(X,Y) = \mu(Y|X)\mu(X),
\end{equation}
and so we have
\begin{equation}
\mu(X|Y) = \frac{\mu(Y | X)\mu(X)}{\mu(Y)}.
\end{equation} |
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