| Version 9 |
Version 8 |
| A \emph{random vector} is a finite-dimensional formal vector of |
A \emph{random vector} is a finite-dimensional formal vector of |
| random variables. The random vector can be written either as a |
random variables. The random vector can be written either as a |
| column or row of random variables, depending on its context and use. |
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 |
So if $X_1,X_2,\ldots,X_n$ are random variables, then |
| $$\textbf{X}=\begin{pmatrix} X_1 \\ X_2 \\ |
$$\textbf{X}=\begin{pmatrix} X_1 \\ X_2 \\ |
| \vdots \\ X_n \end{pmatrix}=\trnsp{(X_1,X_2,\ldots,X_n)}$$ is a |
\vdots \\ X_n \end{pmatrix}=\trnsp{(X_1,X_2,\ldots,X_n)}$$ is a |
| random (column) vector. Similarly, one defines a \emph{random |
random (column) vector. Similarly, one defines a \emph{random |
| matrix} to be a formal matrix whose entries are all random |
matrix} to be a formal matrix whose entries are all random |
| variables. The dimension of a random vector and the dimensions of a |
variables. The dimension of a random vector and the dimensions of a |
| random matrix are assumed to be finite fixed constants. |
random matrix are assumed to be finite fixed constants. |
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| The \emph{distribution of a random vector} |
The \emph{distribution of a random vector} |
| $\textbf{X}=(X_1,X_2,\ldots,X_n)$ is defined to be the joint |
$\textbf{X}=(X_1,X_2,\ldots,X_n)$ is defined to be the joint |
| distribution of its coordinates $X_1,\ldots,X_n$: |
distribution of its coordinates $X_1,\ldots,X_n$: |
| $$F_{\textbf{X}}(\textbf{x}):=F_{X_1,\ldots,X_n}(x_1,\ldots,x_n).$$ |
$$F_{\textbf{X}}(\textbf{x}):=F_{X_1,\ldots,X_n}(x_1,\ldots,x_n).$$ |
| Similary, the \emph{distribution of a random matrix} is the joint |
Similary, the \emph{distribution of a random matrix} is the joint |
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distribution of its matrix components.
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distribution of its matric components.
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| Let $\textbf{X}=(X_1,X_2,\ldots,X_n)$ be a random vector. If |
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 |
$\operatorname{E}[X_i]$ exists ($<\infty$) for each $i$, then the expectation of |
| $\textbf{X}$, called the \emph{mean vector} and denoted by |
$\textbf{X}$, called the \emph{mean vector} and denoted by |
| $\mathbf{E}[\textbf{X}]$, is defined to be: |
$\mathbf{E}[\textbf{X}]$, is defined to be: |
| $$\mathbf{E}[\textbf{X}]:=(\operatorname{E}[X_1],\operatorname{E}[X_2],\ldots, \operatorname{E}[X_n]).$$ |
$$\mathbf{E}[\textbf{X}]:=(\operatorname{E}[X_1],\operatorname{E}[X_2],\ldots, \operatorname{E}[X_n]).$$ |
| Clearly $\mathbf{E}[\textbf{X}]^T=\mathbf{E}[\textbf{X}^T]$. The |
Clearly $\mathbf{E}[\textbf{X}]^T=\mathbf{E}[\textbf{X}^T]$. The |
| expectation of a random matrix is similarly defined. Note that the |
expectation of a random matrix is similarly defined. Note that the |
| definitions of expectations can also be defined via measure theory. Then, |
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. |
using Fubini's Theorem, one can show that the two sets of definitions coincide. |
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| Again, let $\textbf{X}=(X_1,X_2,\ldots,X_n)^T$ be a random vector. |
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 |
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 |
$\operatorname{E}[X_iX_j]$ are defined for all $1\leq i,j \leq n$, then the |
| variance of $\textbf{X}$, denoted by $\textbf{Var}[\textbf{X}]$, is |
variance of $\textbf{X}$, denoted by $\textbf{Var}[\textbf{X}]$, is |
| defined to be: |
defined to be: |
| $$\textbf{Var}[\textbf{X}]:= \mathbf{E}\big[(\textbf{X}-\boldsymbol{\mu})(\textbf{X}-\boldsymbol{\mu})^T\big].$$ |
$$\textbf{Var}[\textbf{X}]:= \mathbf{E}\big[(\textbf{X}-\boldsymbol{\mu})(\textbf{X}-\boldsymbol{\mu})^T\big].$$ |
| It is not hard to see that $\textbf{Var}[\textbf{X}]$ is an $n\times |
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 |
n$ symmetric matrix and it is equal to the covariance matrix of the |
| $X_i$'s. |
$X_i$'s. |
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| \textbf{\PMlinkescapetext{Properties}:} |
\textbf{\PMlinkescapetext{Properties}:} |
| \begin{enumerate} |
\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 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 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 $\boldsymbol{\mu}=\mathbf{E[X]}$, |
\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 |
$\boldsymbol{\Sigma}=\mathbf{Var[X]}$. $\mathbf{A}$ is an $n\times |
| n$ constant matrix. Then |
n$ constant matrix. Then |
| $$\mathbf{E}[\mathbf{X}^T\mathbf{AX}]=\operatorname{tr}(\mathbf{A}\boldsymbol{\Sigma})+ |
$$\mathbf{E}[\mathbf{X}^T\mathbf{AX}]=\operatorname{tr}(\mathbf{A}\boldsymbol{\Sigma})+ |
| \boldsymbol{\mu}^T\mathbf{A}\boldsymbol{\mu}.$$ |
\boldsymbol{\mu}^T\mathbf{A}\boldsymbol{\mu}.$$ |
| \end{enumerate} |
\end{enumerate} |
