Stat 5101 Lecture Notes - School of Statistics

2.5. PROBABILITY THEORY AS LINEAR ALGEBRA 59So we are done, the identification b = h(1) makes the two equations the same.2.5.3 Expectation on Finite Sample SpacesConsider a finite set S and define L 1 to be the set of all real-valued functionson S. This makes L 1 a finite-dimensional vector space. The elements of L 1 differfrom n-tuples only in notation. A random variable X ∈ L 1 is determined by itsvaluesX(s), s ∈ S,and since S is finite, this means X is determined by a finite list of real numbers.If S is indexedS = {s 1 ,...,s n }then we could even, if we wanted, collect these values into an n-tuple(x 1 ,...,x n )wherex i = X(s i ), i =1,...,n,which shows explicitly the correspondence between n-tuples and functions on aset of cardinality n.However, we don’t want to make too much of this correspondence. In factthe only use we want to make of it is the following fact: every linear functionalT on an n-dimensional vector space has the formT (x) =n∑a i x i (2.37)where, as usual, x =(x 1 ,...,x n ). This is sometimes writteni=1T (x) =a ′ xwhere a =(a 1 ,...,a n ) the prime indicating transpose and a and x being consideredas column vectors. Other people writeT (x) =a·xthe operation indicated by the dot being called the scalar product or dot productof the vectors a and x.We now want to change back to our original notation, writing vectors asfunctions on a finite set S rather than n-tuples, in which case (2.37) becomesT (x) = ∑ s∈Sa(s)x(s)Now we want to make another change of notation. If we want to talk aboutvectors that are elements of L 1 (and we do), we should use the usual notation,