TABLE OF CONTENTS Thus far we have emphasized random variables, scalar functions on a sample space that assume real values. In some cases we may wish to model processes or measurements with complex values. Complex outputs can be considered as two-dimensional real vectors with the components being the real and imaginary parts or, equivalently, the magnitude and phase. This special case can be equally well described as a single complex-valued random variable or as a two-dimensional random vector.
More generally, we may have k-dimensional real vector outputs. A random variable is a real-valued function on a sample space (with a technical condition), that is, a function mapping a sample space into the real line 
. The obvious random vector definition is a vector-valued function definition. Under this definition, a random vector is a vector of random variables, a function mapping the sample space into k instead of . Yet even more generally, we may have vectors that are not finite-dimensional, e.g., sequences and waveforms whose values at each time are random variables. This is essentially the definition of a random process. Fundamentally speaking, both random vectors and random processes are simply collections of random variables defined on a common probability space.
Given a probability space ( ? , 
, P), a finite collection of random variables { X i; i =0, 1, , k ?1} is called a random vector. We will often denote a random vector...
