RandomVector 2004-06-30 12:25:09 2007-05-09 15:02:02 Definition random matrix distribution of a random vector distribution of a random matrix mean vector % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb,amscd} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \newcommand{\trnsp}[1]{#1^{\operatorname{T}}} A \emph{random vector} is a finite-dimensional formal vector of random variables. The random vector can be written either as a column or row of random variables, depending on its context and use. So if $X_1,X_2,\ldots,X_n$ are random variables, then $$\textbf{X}=\begin{pmatrix} X_1 \\ X_2 \\ \vdots \\ X_n \end{pmatrix}=\trnsp{(X_1,X_2,\ldots,X_n)}$$ is a random (column) vector. Similarly, one defines a \emph{random matrix} to be a formal matrix whose entries are all random variables. The size of a random vector and the size of a random matrix are assumed to be finite fixed constants. The \emph{distribution of a random vector} $\textbf{X}=(X_1,X_2,\ldots,X_n)$ is defined to be the joint distribution of its coordinates $X_1,\ldots,X_n$: $$F_{\textbf{X}}(\textbf{x}):=F_{X_1,\ldots,X_n}(x_1,\ldots,x_n).$$ Similarly, the \emph{distribution of a random matrix} is the joint distribution of its matrix components. Let $\textbf{X}=(X_1,X_2,\ldots,X_n)$ be a random vector. If $\operatorname{E}[X_i]$ exists ($<\infty$) for each $i$, then the expectation of $\textbf{X}$, called the \emph{mean vector} and denoted by $\mathbf{E}[\textbf{X}]$, is defined to be: $$\mathbf{E}[\textbf{X}]:=(\operatorname{E}[X_1],\operatorname{E}[X_2],\ldots, \operatorname{E}[X_n]).$$ Clearly $\mathbf{E}[\textbf{X}]^T=\mathbf{E}[\textbf{X}^T]$. The expectation of a random matrix is similarly defined. Note that the definitions of expectations can also be defined via measure theory. Then, using Fubini's Theorem, one can show that the two sets of definitions coincide. Again, let $\textbf{X}=(X_1,X_2,\ldots,X_n)^T$ be a random vector. If $\boldsymbol{\mu}$=$\mathbf{E}[\textbf{X}]$ is defined and $\operatorname{E}[X_iX_j]$ are defined for all $1\leq i,j \leq n$, then the variance of $\textbf{X}$, denoted by $\textbf{Var}[\textbf{X}]$, is defined to be: $$\textbf{Var}[\textbf{X}]:= \mathbf{E}\big[(\textbf{X}-\boldsymbol{\mu})(\textbf{X}-\boldsymbol{\mu})^T\big].$$ It is not hard to see that $\textbf{Var}[\textbf{X}]$ is an $n\times n$ symmetric matrix and it is equal to the covariance matrix of the $X_i$'s. \textbf{\PMlinkescapetext{Properties}:} \begin{enumerate} \item If \textbf{X} is an $n$-dimensional random vector with \textbf{A} a $m\times n$ constant matrix and $\boldsymbol{\alpha}$ an $m$-dimensional constant vector, then $$\mathbf{E}[\mathbf{AX}+\boldsymbol{\alpha}]=\mathbf{AE}[\mathbf{X}]+\boldsymbol{\alpha}.$$ \item Same set up as above. Then $$\mathbf{Var}[\mathbf{AX}+\boldsymbol{\alpha}]=\mathbf{AVar}[\mathbf{X}]\mathbf{A}^T.$$ If the ${X_i}$'s are \emph{iid} (independent identically distributed), with variance $\boldsymbol{\sigma}^2$, then $$\mathbf{Var}[\mathbf{AX}+\boldsymbol{\alpha}]=\boldsymbol{\sigma}^2\mathbf{AA}^T.$$ \item Let $\mathbf{X}$ be an $n$-dimensional random vector with $\boldsymbol{\mu}=\mathbf{E[X]}$, $\boldsymbol{\Sigma}=\mathbf{Var[X]}$. $\mathbf{A}$ is an $n\times n$ constant matrix. Then $$\mathbf{E}[\mathbf{X}^T\mathbf{AX}]=\operatorname{tr}(\mathbf{A}\boldsymbol{\Sigma})+ \boldsymbol{\mu}^T\mathbf{A}\boldsymbol{\mu}.$$ \end{enumerate}