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Random Vectors

Module by: Don Johnson. E-mail the author

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

EX=xp X xdx=E X 1 E X L X x x p X x X 1 X L
(1)
The covariance matrix K X K X is an L × L L × L matrix consisting of all possible covariances among the random vector's components.
i,j,ij1L: K X i,j=cov X i X j =E X i X j ¯E X i E X j ¯ i j i j 1 L K X i j cov X i X j X i X j X i X j
(2)
Using matrix notation, the covariance matrix can be written as K X =E(XEX)(XEX)T 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 no zero-variance component, its covariance matrix is positive-definite. Note in particular that when the random variables are real-valued, the diagonal elements of a covariance matrix equal the variances of the components: K X i,i= σ X i 2 K X i i σ X i 2 . Circular random vectors are complex-valued with uncorrelated, identically distributed, real and imaginary parts. In this case, E| X i |2=2 σ X i 2 X i 2 2 σ X i 2 , and E X i 2=0 X i 2 0 . By convention, σ X i 2 σ X i 2 denotes the variance of the real (or imaginary) parts. The characteristic function of a real-valued random vector is defined to be
Φ X iν=EeiνTX Φ X ν ν X
(3)

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

PrmaxX<μ=P X μμ X μ P X μ μ
(4)
Assuming that the components of XX are statistically independent, this expression becomes
PrmaxX<μ=i=1dimXP X i μ X μ i 1 dim X P X i μ
(5)

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