Vector space Interpretation of Random Variables

      The interpretation of random variables as elements of a vector space helps in understanding many operations involving random variables. We start with an introduction to the concepts of the vector space. We will discuss the principles of the minimum mean-square error estimation and the linear minimum mean-square error estimation of a signal from noise and the vector-space interpretation of the later.

   Vector space

           Consider a set with elements called vectors and the field of real numbers .is called a vector space if and only if

1. An operation vector addition '+' is defined in such that ( , +) is a commutative group. Thus ( , +) satisfies the following properties.
   
   
   1. For any pair of elements , there exists a unique element .
   
   
   
   2. Vector addition is associative: for any three vectors