Random Vectors 1.2 2003/05/14 2003/08/08 14:00:13.428 GMT-5 Don Johnson dhj@rice.edu Don Johnson dhj@rice.edu Eileen Krause erkrause@rice.edu Kyle Clarkson kclarks@rice.edu Elizabeth Gregory lizzardg@rice.edu Kevin Duh kevinduh@rice.edu Mariyah Poonawala mariyah@rice.edu Jeffrey Silverman jsilv@rice.edu 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 variables considered 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 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 . Circular random vectors are complex-valued with uncorrelated, identically distributed, real and imaginary parts. In this case, X i 2 2 σ X i 2 , and X i 2 0 . By convention, 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 ν ν X 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 μ is equal to the probability that all of the components are less than μ. X μ P X μ … μ Assuming that the components of X are statistically independent, this expression becomes X μ i 1 dim X P X i μ