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1.3 Random vectors

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A random vector X is an ordered sequence of random variables X X 1 X L . The density function of a random vector is defined in a manner similar to that for pairs of random variablesconsidered previously. The expected value of a random vector is the vector of expected values.

X x ∞ ∞ x p X x X 1 X L
The covariance matrix K X is an L L matrix consisting of all possible covariances among the random vector's components.
i j i j 1 L K X i j cov X i X j X i X j X i X j
Using matrix notation, the covariance matrix can be written as K X X X X X . Using this expression, the covariance matrix is seen to be a symmetric matrix and, when the random vector has nozero-variance component, its covariance matrix is positive-definite. Note in particular that when the randomvariables are real-valued, the diagonal elements of a covariance matrix equal the variances of the components: K X i i X i 2 . Circular random vectors are complex-valued with uncorrelated, identically distributed, realand imaginary parts. In this case, X i 2 2 X i 2 , and X i 2 0 . By convention, X i 2 denotes the variance of the real (or imaginary) parts. The characteristic function of a real-valued random vector isdefined to be
X X

The maximum of a random vector is a random variables whose probability density is usually quite different from thedistributions of the vector's components. The probability that the maximum is less than some number is equal to the probability that all of the components are less than .

X P X
Assuming that the components of X are statistically independent, this expression becomes
X i 1 dim X P X i

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Source:  OpenStax, Statistical signal processing. OpenStax CNX. Dec 05, 2011 Download for free at http://cnx.org/content/col11382/1.1
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