Random Vectors

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Vectors :: Covariance :: Functions :: Application :: Markov Model

Random Vectors

Random vectors are an extension of the bivariate random variables.

r.v.s $X_1, X_2, \ldots, X_n$ define a measurable mapping from an underlying sample space $(\Omega, \Fc)$ to $(\Rbb^n,\Bc^n)$ , where $\Bc^n$ is the smallest $\sigma$ -field containing all sets of the form

$\begin{displaymath}\{(x_1,x_2,\ldots,x_n): a_1 < x_1 \leq b_1, a_2 < x_2 \leq b_2, \cdots, a_n < x_n \leq b_n\}. \end{displaymath}$ < x_1 \leq b_1, a_2 < x_2 \leq b_2, \cdots, a_n < x_n \leq b_n\}. \end{displaymath}" border="0" height="38" width="452" />

$\begin{definition} The {\bf joint distribution} of $X_1, \ldots, X_n$ is \begi... ...c \end{displaymath}This probability is denoted as $P_\Xbf(B)$. \end{definition}$

$\begin{definition} The {\bf joint cumulative distribution function} c.d.f. if \... ... a_n) = F_\Xbf(\abf), \qquad \abf \in \Rbb^n. \end{displaymath}\end{definition}$

$\begin{definition} The {\bf joint probability mass function} (p.m.f.) is \begin{... ...th}p_\Xbf(\abf) = P(X_1=a_1,\ldots, X_n = a_n) \end{displaymath}\end{definition}$

$\begin{definition} The {\bf joint probability density function} (p.d.f.) $f_\Xb... ..._2 \cdots dx_n \end{displaymath}for a continuous random vector. \end{definition}$
Fact: $X_1, X_2, \ldots, X_n$ are independent if $F_\Xbf$ or $p_\Xbf$ or $f_\Xbf$ factor into products of marginals.

Suppose $g:(\Rbb^n,\Bc^n) \rightarrow (\Rbb^n,\Bc^n)$ is measurable. Then $g(X_1,\ldots, X_n)$ is a random variable. Law of unconscious statistician:

$\begin{displaymath}E[g(X_1,\ldots, X_n)] = \begin{cases} \int \cdots \int g(x_... ...i_1}, x_{i_2}, \ldots, x_{i_n}) & \text{discrete}. \end{cases}\end{displaymath}$

Copyright 2008, Todd Moon. Cite/attribute Resource. admin. (2006, May 31). Random Vectors. Retrieved November 23, 2009, from Free Online Course Materials — USU OpenCourseWare Web site: http://ocw.usu.edu/Electrical_and_Computer_Engineering/Stochastic_Processes/lec3_1.html. This work is licensed under a Creative Commons License.

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	Info Random Vectors Document Actions Vectors :: Covariance :: Functions :: Application :: Markov Model Random Vectors Random vectors are an extension of the bivariate random variables. r.v.s $X_1, X_2, \ldots, X_n$ define a measurable mapping from an underlying sample space $(\Omega, \Fc)$ to $(\Rbb^n,\Bc^n)$ , where $\Bc^n$ is the smallest $\sigma$ -field containing all sets of the form $\begin{displaymath}\{(x_1,x_2,\ldots,x_n): a_1 < x_1 \leq b_1, a_2 < x_2 \leq b_2, \cdots, a_n < x_n \leq b_n\}. \end{displaymath}$ < x_1 \leq b_1, a_2 < x_2 \leq b_2, \cdots, a_n < x_n \leq b_n\}. \end{displaymath}" border="0" height="38" width="452" /> $\begin{definition} The {\bf joint distribution} of $X_1, \ldots, X_n$ is \begi... ...c \end{displaymath}This probability is denoted as $P_\Xbf(B)$. \end{definition}$ $\begin{definition} The {\bf joint cumulative distribution function} c.d.f. if \... ... a_n) = F_\Xbf(\abf), \qquad \abf \in \Rbb^n. \end{displaymath}\end{definition}$ $\begin{definition} The {\bf joint probability mass function} (p.m.f.) is \begin{... ...th}p_\Xbf(\abf) = P(X_1=a_1,\ldots, X_n = a_n) \end{displaymath}\end{definition}$ $\begin{definition} The {\bf joint probability density function} (p.d.f.) $f_\Xb... ..._2 \cdots dx_n \end{displaymath}for a continuous random vector. \end{definition}$ Fact: $X_1, X_2, \ldots, X_n$ are independent if $F_\Xbf$ or $p_\Xbf$ or $f_\Xbf$ factor into products of marginals. Suppose $g:(\Rbb^n,\Bc^n) \rightarrow (\Rbb^n,\Bc^n)$ is measurable. Then $g(X_1,\ldots, X_n)$ is a random variable. Law of unconscious statistician: $\begin{displaymath}E[g(X_1,\ldots, X_n)] = \begin{cases} \int \cdots \int g(x_... ...i_1}, x_{i_2}, \ldots, x_{i_n}) & \text{discrete}. \end{cases}\end{displaymath}$ Copyright 2008, Todd Moon. Cite/attribute Resource. admin. (2006, May 31). Random Vectors. Retrieved November 23, 2009, from Free Online Course Materials — USU OpenCourseWare Web site: http://ocw.usu.edu/Electrical_and_Computer_Engineering/Stochastic_Processes/lec3_1.html. This work is licensed under a Creative Commons License.	Reuse Course Download this course

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