Difference between revisions of "ECE600 F13 Random Vectors mhossain" - Rhea

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Random Vectors

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

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
  

and the probability density function of X is

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

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
  

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

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

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

and the covariance matrix C_X as

The mean vector $ \mu $_X is

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

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

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

be a scalar random variable. Then

So

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

where

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

Then

If X$ _1 $,..., X$ _n $ are independent, then

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

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

is a Gaussian random variable ∀[a$ _0 $,..., a$ _n $] ∈ R$ ^{n+1} $.

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

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

Proof $ \qquad $ Let

for

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

where

and

where

and C_X is the covariance matrix of X.

Now

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

Note that we can use the equation

to show that if X is Gaussian, then 

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

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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