5. Free Probability The definition of free probability is analogous to the concept of independence in classical probability.The definition of freeness is as follows: a)The free probability developed as a probability theory can be applied for noncommutative random variables like matrices; b)These random variables are the elements in a non-commutative probability space that can be defined by a pair (A,φ) where A is a unital algebra,and o is a normalized linear functional on A.In random matrix backgound,A is a IXI matrix and o is the normalized monent for the matrix,which is defined as 44 5. Free Probability The definition of free probability is analogous to the concept of independence in classical probability. The definition of freeness is as follows: a)The free probability developed as a probability theory can be applied for noncommutative random variables like matrices; b)These random variables are the elements in a non-commutative probability space that can be defined by a pair ( , ) A  where A is a unital algebra, and φ is a normalized linear functional on A. In random matrix backgound, A is a l×l matrix and φ is the normalized monent for the matrix, which is defined as 1 ( ) ( ) tr l   A A