Stat 5101 Lecture Notes - School of Statistics

2.5. PROBABILITY THEORY AS LINEAR ALGEBRA 57you may be wondering what the connection between random variables and vectorscould possibly be. Random variables are functions (on the sample space)and functions aren’t n-tuples or matrices.But n-tuples are functions. You just have to change notation to see it. Writex(i) instead of x i , and it’s clear that n-tuples are a special case of the functionconcept. An n-tuple is a function that maps the index i to the value x i .So the problem here is an insufficiently general notion of vectors. You shouldthink of functions (rather than n-tuples or matrices) as the most general notionof vectors. Functions can be added. If f and g are functions on the samedomain, then h = f + g meansh(s) =f(s)+g(s),for all s in the domain.Functions can be multiplied by scalars. If f is a function and a is a scalar (realnumber), then h = af meansh(s) =af(s),for all s in the domain.Thus the set of scalar-valued functions on a common domain form a vector space.In particular, the scalar-valued random variables of a probability model (all realvaluedfunctions on the sample space) form a vector space. Theorem 2.25 assertsthat L 1 is a subspace of this vector space.Linear Transformations and Linear FunctionalsIf U and V are vector spaces and T is a function from U to V , then we saythat T is linear ifT (ax + by) =aT (x)+bT (y),for all vectors x and y and scalars a and b. (2.35)Such a T is sometimes called a linear transformation or a linear mapping ratherthan a linear function.The set of scalars (the real numbers) can also be thought of as a (onedimensional)vector space, because scalars can be added and multiplied byscalars. Thus we can also talk about scalar-valued (real-valued) linear functionson a vector space. Such a function satisfies the same property (2.35). The onlydifference is that it is scalar-valued rather than vector-valued. In linear algebra,a scalar-valued linear function is given the special name linear functional.Theorem 2.25 asserts that the mapping from random variables X to theirexpectations E(X) is a linear functional on L 1 . To understand this you have tothink of E as a function, a rule that assigns values E(X) to elements X of L 1 .Proof of Theorem 2.25. The existence assertions of Properties E1 and E2 assertthat random variables in L 1 can be added and multiplied by scalars yielding aresult in L 1 .ThusL 1 is a vector space. Property 2.1 now says the same thingas (2.35) in different notation. The map E, being scalar-valued, is thus a linearfunctional on L 1 .