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 $X_1,X_2,\mathrm{\dots },X_n$ are random variables, then

$$\text{𝐗}=\left(\begin{array}{c}\hfill X_1\hfill \\ \hfill X_2\hfill \\ \hfill \mathrm{⋮}\hfill \\ \hfill X_n\hfill \end{array}\right)={(X_1,X_2,\mathrm{\dots },X_n)}^{\mathrm{T}}$$

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 $\text{𝐗}=(X_1,X_2,\mathrm{\dots },X_n)$ is defined to be the joint distribution of its coordinates $X_1,\mathrm{\dots },X_n$:

$$F_{\text{𝐗}}(\text{𝐱}):=F_{X_1,\mathrm{\dots },X_n}(x_1,\mathrm{\dots },x_n).$$

Similarly, the distribution of a random matrix is the joint distribution of its matrix components.

Let $\text{𝐗}=(X_1,X_2,\mathrm{\dots },X_n)$ be a random vector. If $\mathrm{E}[X_i]$ exists () for each $i$, then the expectation of X, called the mean vector and denoted by $𝐄[\text{𝐗}]$, is defined to be:

$$𝐄[\text{𝐗}]:=(\mathrm{E}[X_1],\mathrm{E}[X_2],\mathrm{\dots },\mathrm{E}[X_n]).$$

Clearly $𝐄{[\text{𝐗}]}^T=𝐄[{\text{𝐗}}^T]$. 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 $\text{𝐗}={(X_1,X_2,\mathrm{\dots },X_n)}^T$ be a random vector. If $𝝁$=$𝐄[\text{𝐗}]$ is defined and $\mathrm{E}[X_i{X}_j]$ are defined for all $1\le i,j\le n$, then the variance of X, denoted by $\text{𝐕𝐚𝐫}[\text{𝐗}]$, is defined to be:

$$\text{𝐕𝐚𝐫}[\text{𝐗}]:=𝐄\left[(\text{𝐗}-𝝁){(\text{𝐗}-𝝁)}^T\right].$$

It is not hard to see that $\text{𝐕𝐚𝐫}[\text{𝐗}]$ is an $n\times n$ symmetric matrix and it is equal to the covariance matrix of the $X_i$’s.

:

1. If X is an $n$-dimensional random vector with A a $m\times n$ constant matrix and $𝜶$ an $m$-dimensional constant vector, then

   $$𝐄[\mathrm{𝐀𝐗}+𝜶]=\mathrm{𝐀𝐄}[𝐗]+𝜶.$$

2. Same set up as above. Then

   $$\mathrm{𝐕𝐚𝐫}[\mathrm{𝐀𝐗}+𝜶]=\mathrm{𝐀𝐕𝐚𝐫}[𝐗]{𝐀}^T.$$

   If the $X_i$’s are iid (independent identically distributed), with variance ${𝝈}^2$, then

   $$\mathrm{𝐕𝐚𝐫}[\mathrm{𝐀𝐗}+𝜶]={𝝈}^2{\mathrm{𝐀𝐀}}^T.$$

3. Let $𝐗$ be an $n$-dimensional random vector with $𝝁=𝐄[𝐗]$, $𝚺=\mathrm{𝐕𝐚𝐫}[𝐗]$. $𝐀$ is an $n\times n$ constant matrix. Then

   $$𝐄[{𝐗}^T\mathrm{𝐀𝐗}]=\mathrm{tr}(𝐀𝚺)+{𝝁}^T𝐀𝝁.$$

Title	random vector
Canonical name	RandomVector
Date of creation	2013-03-22 14:27:20
Last modified on	2013-03-22 14:27:20
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	15
Author	CWoo (3771)
Entry type	Definition
Classification	msc 62H99
Classification	msc 15A52
Defines	random matrix
Defines	distribution of a random vector
Defines	distribution of a random matrix
Defines	mean vector