# random vector

A 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

 $\textbf{X}=\begin{pmatrix}X_{1}\\ X_{2}\\ \vdots\\ X_{n}\end{pmatrix}=(X_{1},X_{2},\ldots,X_{n})^{\operatorname{T}}$

is a random (column) vector. Similarly, one defines a 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 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 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 X, called the 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 Fubiniโ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 $\operatorname{E}[X_{i}X_{j}]$ are defined for all $1\leq i,j\leq n$, then the variance of 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 $X_{i}$โs.

:

1. 1.

If X is an $n$-dimensional random vector with A a $m\times n$ constant matrix and $\boldsymbol{\alpha}$ an $m$-dimensional constant vector, then

 $\mathbf{E}[\mathbf{AX}+\boldsymbol{\alpha}]=\mathbf{AE}[\mathbf{X}]+% \boldsymbol{\alpha}.$
2. 2.

Same set up as above. Then

 $\mathbf{Var}[\mathbf{AX}+\boldsymbol{\alpha}]=\mathbf{AVar}[\mathbf{X}]\mathbf% {A}^{T}.$

If the ${X_{i}}$โs are iid (independent identically distributed), with variance $\boldsymbol{\sigma}^{2}$, then

 $\mathbf{Var}[\mathbf{AX}+\boldsymbol{\alpha}]=\boldsymbol{\sigma}^{2}\mathbf{% AA}^{T}.$
3. 3.

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

 $\mathbf{E}[\mathbf{X}^{T}\mathbf{AX}]=\operatorname{tr}(\mathbf{A}\boldsymbol{% \Sigma})+\boldsymbol{\mu}^{T}\mathbf{A}\boldsymbol{\mu}.$
Title random vector RandomVector 2013-03-22 14:27:20 2013-03-22 14:27:20 CWoo (3771) CWoo (3771) 15 CWoo (3771) Definition msc 62H99 msc 15A52 random matrix distribution of a random vector distribution of a random matrix mean vector