Stat 5101 Lecture Notes - School of Statistics

60 Stat 5101 (Geyer) Course Notesdenoting those elements (which are random variables) by X rather than x andtheir components by X(s) givingT (X) = ∑ s∈Sa(s)X(s). (2.38)To summarize the argument of this section so farTheorem 2.26. For probability models on a finite sample space S, every linearfunctional on L 1 has the form (2.38).But not every linear functional is an expectation operator. Every linearfunctional satisfies two of the probability axioms (homogeneity and additivity).But a linear functional need not satisfy the other two (positivity and norm).In order that (2.38) be positive whenever X ≥ 0, that is, when X(s) ≥ 0,for all s, it is required thata(s) ≥ 0, s ∈ S. (2.39a)In order that (2.38) satisfy the norm property (2.4) it is required that∑a(s) =1,s∈S(2.39b)because X = 1 means X(s) = 1, for all s. We have met functions like thisbefore: a function a satisfying (2.39a) and (2.39b) we call a probability density.Lindgren calls them probability functions (p. f.’s).Theorem 2.27. For probability models on a finite sample space S, every expectationoperator on L 1 has the formE(X) = ∑ s∈Sp(s)X(s) (2.40)for some function p : S → R satisfyingp(s) ≥ 0, s ∈ S, (2.41a)and∑p(s) =1.s∈S(2.41b)A function p as defined in the theorem is called a probability density or justa density.Remark. Theorem 2.27 is also true if the word “finite” in the first sentence isreplaced by “countable” (see Theorem 2.30).