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4
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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: |
2005-11-27 10:20:33 |
| Classification: |
msc:60A99 |
Revision comment (for changes between this and next version):
| If $\mu(Y)=0$, then $\mu(X|Y)$ is not defined. |
Preamble:
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
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\newcommand{\mv}[1]{\mathbf{#1}} % matrix or vector
\newcommand{\mvt}[1]{\mv{#1}^{\mathrm{T}}}
\newcommand{\mvi}[1]{\mv{#1}^{-1}}
\newcommand{\mderiv}[1]{\frac{\md}{\md {#1}}} %d/dx
\newcommand{\mnthderiv}[2]{\frac{\md^{#2}}{\md {#1}^{#2}}} %d^n/dx
\newcommand{\mpderiv}[1]{\frac{\partial}{\partial {#1}}} %partial d^n/dx
\newcommand{\mnthpderiv}[2]{\frac{\partial^{#2}}{\partial {#1}^{#2}}} %partial d^n/dx
\newcommand{\borel}{\mathfrak{B}}
\newcommand{\integers}{\mathbb{Z}}
\newcommand{\rationals}{\mathbb{Q}}
\newcommand{\reals}{\mathbb{R}}
\newcommand{\complexes}{\mathbb{C}}
\newcommand{\naturals}{\mathbb{N}}
\newcommand{\defined}{:=}
\newcommand{\var}{\mathrm{var}}
\newcommand{\cov}{\mathrm{cov}}
\newcommand{\corr}{\mathrm{corr}}
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\newcommand{\abs}[1]{\left|#1\right|}
\newcommand{\norm}[1]{\left|\left|#1\right|\right|}
\newcommand{\esssup}{\mathrm{ess\ sup}}
\newcommand{\Lspace}[1]{L^{#1}}
\newcommand{\Lone}{\Lspace{1}}
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\newcommand{\Lp}{\Lspace{p}}
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\newcommand{\Linf}{\Lspace{\infty}} |
Content:
Let $(\Omega, \borel, \mu)$ be a probability space, and let $X,Y\in\borel$ be events.
The \emph{conditional probability} of $X$ given $Y$ is defined as
\begin{equation}
\mu(X|Y) = \frac{\mu(X \cap Y)}{\mu(Y)}
\end{equation}
provided $\mu(Y)>0$.
If $\mu(X)>0$ and $\mu(Y)>0$, then
\begin{equation}
\mu(X|Y)\mu(Y) = \mu(X\cap Y) = \mu(Y|X)\mu(X),
\end{equation}
and so also
\begin{equation}
\mu(X|Y) = \frac{\mu(Y | X)\mu(X)}{\mu(Y)},
\end{equation}
which is Bayes' Theorem. |
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