random vector

A random vector is a finite-dimensional formal vector ofrandom variables. The random vector can be written either as acolumn 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 arandom (column) vector. Similarly, one defines a randommatrix to be a formal matrix whose entries are all randomvariables. The size of a random vector and the size of arandom 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 jointdistribution 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 jointdistribution 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 ofX, 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 $𝐄{[\text{𝐗}]}^T=𝐄[{\text{𝐗}}^T]$. Theexpectation of a random matrix is similarly defined. Note that thedefinitions 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 thevariance of X, denoted by $\text{𝐕𝐚𝐫}[\text{𝐗}]$, isdefined 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

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