ECE600 F13 Random Vectors mhossain - Rhea

Random Variables and Signals

Topic 17: Random Vectors

Random Vectors

Definition $\qquad$ let X $$ _1 $$ ,..., X $$ _n $$ be n random variables on (S,F,P). The column vector X is given by

\underline{X} = [X_1,...,X_n]^T

is a random vector (RV) on (S,F,P).

Fig 1: The mapping from the sample space to the event space under X

$ _j $

.

We can view X( $\omega$ ) as a point in R $$ ^n $$ ∀ $\omega$ ∈ S.
Much of what we need to work with random vectors we can get by a simple extension of what we have developed for n = 2.

For example:

The cumulative distribution function of X is

F_{\underline X}(\underline x) = P(X_1\leq x_1,...,X_n\leq x_n)\;\;\forall\underline x = [x_1,...,x_n]^T\in\mathbb R^n

and the probability density function of X is

f_{\underline X}(\underline x) = \frac{\partial^nF_{\underline X}(\underline x)}{\partial x_1...\partial x_n}

For any D ⊂ R $$ ^n $$ such that D ∈ B(R $$ ^2 $$ ),

P(\underline X \in D) = \int_D f_{\underline X}(\underline x)d\underline x

Note that B(R

$ ^n $

) is the

\sigma

-field generated by the collection of all open n-dimensional hypercubes (more formally, k-cells) in R

$ ^n $

.

The formula for the joint pdf of two functions of two random variables can be extended to find the pdf of n functions of n random variables (see Papoulis).

The random variables X $$ _1 $$ ,..., X $$ _n $$ are statistically independent if the events {X $$ _1 $$ ∈ A $$ _1 $$ },..., {X $$ _n $$ ∈ A $$ _n $$ } are independent ∀A $$ _1 $$ , ..., A $$ _n $$ ∈ B(R). An equivalent definition is that X $$ _1 $$ ,..., X $$ _n $$ are independent if

f_{\underline X}(\underline x)=\prod_{i=1}^nf_{X_i}(x_i)\;\;\forall\underline x\in\mathbb R^n

Random Vectors: Moments

We will spend some time on moments of random vectors. We will be especially interested in pairwise covariances/correlations.
The correlation between X $$ _j $$ and X $$ _k $$ is denoted R $_{jk}$ , so

R_{jk} \equiv E[X_jX_k]

and the covariance is C $_{jk}$ :

C_{jk}\equiv E[(X_j-\mu_{X_j})(X_k-\mu_{X_k})]

For a random vector X, we define the correlation matrix R_X as

R_{\underline X}=\begin{bmatrix} R_{11} & \cdots & R_{1n} \\ \vdots & & \vdots \\ R_{n1} & \cdots & R_{nn} \end{bmatrix}

and the covariance matrix C_X as

C_{\underline X}=\begin{bmatrix} C_{11} & \cdots & C_{1n} \\ \vdots & & \vdots \\ C_{n1} & \cdots & C_{nn} \end{bmatrix}

The mean vector $\mu$ _X is

\mu_{\underline X}=[\overline X_1,\cdots,\overline X_n]^T

Note that the correlation matrix and the covariance matrix can be written as

\begin{align} R_{\underline X}&=E[\underline X\underline X^T] \\ C_{\underline X}&=E[(\underline X-\mu_{\underline X})(\underline X-\mu_{\underline X})^T] \end{align}

Note that $\mu$ _X, R_X and C_X are the moments we most commonly use for the random vectors.

We need to discuss an important property of R_X, but first, a definition from Linear Algebra.

Definition $\qquad$ An n × m matrix B with b $_{ij}$ as its i,j $^{th}$ entry is non-negative definite (NND) (or positive semidefinite) if

\sum_{i=1}^n\sum_{j=1}^n x_i x_j b_{ij} \geq 0

for all real vectors [x $$ _1 $$ ,...,x $$ _n $$ ] ∈ R $$ ^n $$ .

That is to say that for any real vector x, the product x $$ ^T $$ Ax, where A is a real matrix, is non negative.

Theorem $\qquad$ For any random vector X, R_X is NND.

Proof: $\qquad$ let a be an arbitrary real vector in R $$ ^n $$ , and let

Y = \underline a^T\underline X = \underline X^T\underline a

be a scalar random variable. Then

\begin{align}0\leq E[Y^2]&=E[\underline a^T\underline X\underline X^T\underline a] \\ &=\underline a^TE[\underline X\underline X^T]\underline a \\ &=\underline a^TR_{\underline X}\underline a \end{align}

So

0\leq \underline a^TR_{\underline X}\underline a = \sum_{i=1}^n\sum_{j=1}^na_ia_jR_{ij}

and thus, R_X is NND

Note: C_X is also NND.

Characteristic Functions of Random Vectors

Definition $\qquad$ let X be a random vector on (S,F,P). Then the characteristic function of X is

\Phi_{\underline X}(\underline\Omega)=E\left[e^{i\sum_{j=1}^n\omega_jX_j}\right]

where

\underline\Omega = [\omega_1,...,\omega_n]^T \in \mathbb R^n

The characteristic function Φ_X is extremely useful for finding pdfs of sums of random variables.
Let

Z=\sum_{j=1}^nX_j

Then

\Phi_Z(\omega)=E\left[e^{i\omega\sum_{j=1}^nX_j}\right] = \Phi_{\underline X}(\omega,...,\omega)

If X $$ _1 $$ ,..., X $$ _n $$ are independent, the n

\Phi_Z(\omega)=\prod_{j=1}^n\Phi_{X_j}(\omega)

If, in addition, X $$ _1 $$ ,..., X $$ _n $$ are identically distributed with common characteristic function Φ $$ _X $$ , then

\Phi_Z(\omega) = (\Phi_X(\omega))^n \

Gaussian Random Vectors

Definition $\qquad$ Let X be a random vector on (S,F,P). Then X is Gaussian and X $$ _1 $$ ,..., X $$ _n $$ are said to be jointly Gaussian iff

Z=a_0+\sum_{j=1}^na_jX_j

is a Gaussian random variable ∀[a $$ _1 $$ ,..., a $$ _n $$ ] ∈ R $^{n+1}$ .

Now we will show that the characteristic function of a Gaussian random vector X is

\Phi_{\underline X}(\underline\Omega) = e^{i\underline\Omega^T\mu_{\underline X}-\frac{1}{2}\underline\Omega^TC_{\underline X}\underline\Omega^T}

where $\mu$ _X is the mean vector of X and C_X is the covariance matrix.

Proof $\qquad$ Let

Z = \sum_{j=1}^n\omega_jX_j

for

\underline\Omega \in \mathbb R^n

Then Z is a Gaussian random variable since X is Gaussian. So

\Phi_Z(\omega)=e^{i\omega\mu_Z}e^{-\frac{1}{2}\omega^2\phi_Z^2}

where

\begin{align} \mu_Z&=E\left[\sum_{j=1}^n\omega_jX_j\right] \\ &=\sum_{j=1}^n\omega_j\mu_{X_j} \\ &=\underline\Omega^T\mu_{\underline X} \end{align}

and

\begin{align} \sigma_Z^2 &= \mbox{Var}\left(\sum_{j=1}^n\omega_jX_j\right) \\ &= \sum_{j=1}^n\sum_{k=1}^n\sigma_{jk}\omega_j\omega_k \\ &=\underline\Omega^TC_{\underline X}\underline\Omega \end{align}

where

\sigma_{jk}^2 = E[(X_j-\mu_{X_j})(X_k-\mu_{X_k})]

and C_X is the covariance matrix of X.

Now

\begin{align} \Phi_{\underline X}(\omega)&= E\left[e^{i\sum_{j=i}^n\omega_jX_j}\right] \\ &= \Phi_Z(\omega) \end{align}

Plugging the expressions for $\mu_Z$ and $\sigma_Z$ $$ ^2 $$ into Φ $$ _Z $$ gives

\Phi_{\underline X}(\omega) =e^{i\underline\Omega^T\mu_{\underline X}}e^{-\frac{1}{2}\underline\Omega^TC_{\underline X}\underline\Omega}

Note that we can use the equation

f_{\underline X}(\underline x) = \frac{1}{(2\pi)^n}\int_{\mathbb R^n}\Phi_{\underline X}(\underline\Omega)e^{-i\underline\Omega^T\underline x}d\underline\Omega

to show that if X is Gaussian, then

f_{\underline X}(\underline x) = \frac{1}{\sqrt{(2\pi)^n|C_{\underline X}|}}e^{-\frac{1}{2}(\underline X-\mu_{\underline X})^TC_{\underline X}^{-1}(\underline X-\mu_{\underline X})}

\forall \underline x\in\mathbb R^n

Note that if X $$ _1 $$ ,..., X $$ _n $$ are pairwise uncorrelated, then C_X is the covariance matrix of X is diagonal and

f_{\underline X}(\underline x) = \frac{e^{-\frac{1}{2}\sum_{j=1}^n\frac{x_j^2}{\sigma_j^2}}}{\sqrt{(2\pi)^n\prod_{j=1}^n\sigma_j^2}}

Then, we can find the joint characteristic function and the joint pdf of the jointly Gaussian random variables X and Y using the forms for a Gaussian random vector with n = 2, X $$ _! $$ = X and X $$ _2 $$ = Y.

References

M. Comer. ECE 600. Class Lecture. Random Variables and Signals. Faculty of Electrical Engineering, Purdue University. Fall 2013.

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ECE600 F13 Random Vectors mhossain - Rhea

Contents

Random Vectors

Random Vectors: Moments

Characteristic Functions of Random Vectors

Gaussian Random Vectors

References

Questions and comments

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