Slides09-2010

# Slides09-2010 -...

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\documentclass{beamer} \ \usepackage{beamerthemesplit} \usetheme{Boadilla} \usecolortheme{albatross} \ \title[Moments of Multiple Random Variables]{Moments of More that One Random Variable: Lecture IX} \author{Charles B. Moss} \date{\today} \ \begin{document} \ \frame{\titlepage} \ \section[Outline]{} \frame{\tableofcontents} \ \section{Covariance and Correlation} \frame { \frametitle{Covariance and Correlation} \begin{itemize} \item {\bf Definition 4.3.1} The covariance between two random variables $X$ and $Y$ can be defined as $\begin{array}{c} \displaystyle {\rm Cov}\left( X,Y \right) = {\rm E} \left[ \left( X - {\rm E} \left[ X \right] \right) \left( Y - {\rm E} \left[ Y \right] \right) \right] \\ \displaystyle \; = {\rm E} \left[ XY - X {\rm E} \left[ Y \right] - {\rm E} \left[ x \right] Y + {\rm E} \left[ X \right] \left[ Y \right] \right] \\ \displaystyle \; = {\rm E} \left[ XY \right] - {\rm E} \left[ X \right] {\rm E} \left[Y \right] - {\rm E} \left[ X \right] {\rm E} \left[ Y \right] + {\rm E} \left[ X \right] {\rm E} \left[ Y \right] \\ \displaystyle \; = {\rm E} \left[ XY \right] - {\rm E} \left[ X \right] {\rm E }\left[ Y \right] \end{array} \label{eqn:lect09-001} \ \begin{itemize} \item Note that this is simply a generalization of the standard variance formulation. Specifically, letting$Y \rightarrow X$yields f \begin{array}{c} \displaystyle {\rm Cov}\left( XX \right) = {\rm E} \left[ XX \right] - {\rm E} \left[ X \right] {\rm E} \left[ X \right] \\ \displaystyle \; = {\rm E} \left[ X^2 \right] - \left( {\rm E} \left[ X \right] \right)^2 \end{array} \label{eqn:lect09-002} \ \item From a sample perspective, we have \begin{array}{c} This preview has intentionally blurred sections. Sign up to view the full version. View Full Document V \left( X \right) = \frac{1}{N} \sum_{i=1}^N x_i^2 - \bar x ^2 \\ \\ {\rm Cov}\left( X, Y \right) = \frac{1}{N} \sum_{i=1}^N x_i y_i - \bar x \bar y \end{array} \label{eqn:lect09-003} \ \end{itemize} \end{itemize} } \frame { \begin{itemize} \item Continued \begin{itemize} \item Together the variance and covariance matrices are typically written as a variance matrix a \Sigma = \left[ \begin{array}{cc} V\left( X \right) & {\rm Cov} \left( X,Y \right) \\ {\rm Cov} \left( Y,X \right) & V \left( Y \right) \end{array} \right] end{array} \right] \label{eqn:lect09-004} \ \noindent Note that${\rm Cov}\left( X,Y \right) = \sigma_{xy} = \sigma_{yx} = {\rm Cov}\left( Y,X \right)\$. \item Substituting the sample measures into the variance matrix yields \begin{array}{c} S = \left[ \begin{array}{cc} s_{xx} & s_{xy} \\ s_{yx} & s_{yy} \end{array} \right] = \left[ \begin{array}{cc} \frac{1}{N} \sum_{i=1}^N x_i x_i - \bar x \bar x & \frac{1}{N} \sum_{i=1}^N x_i y_i \bar x \bar y \\ \frac{1}{N} \sum_{i=1}^N y_i x_i - \bar y \bar x & \frac{1}{N} \sum_{i=1}^N y_i y_i - \bar y \bar y \end{array} \right] \\ \\
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