How do we decide that some function X(:S-R is a random variables relative to a given a-field F? From the discussion of the a-field o()generated by the set J=Br: a E R where B2=(oo, c we know that B= a(J) and if X( is such that X-1(-∞,x])={s:X(s)∈(-∞,x,s∈S}∈ F for all(-∞,x∈B then X-(B)={s:X(s)∈B,s∈S}∈ F for all B∈B In other words. when we want to establish that x is a random variables or define Pr( we have to look no further than the half-closed interval(oo, a],and the a-field o() they generate, whatever the range Rr. Let us use the shorthand notation{X(s)≤} instead of{s:X(s)∈(-∞,r,s∈S} to the above examp X-1(-∞,x])={s:X(s)≤x} {(TT)} X≤0( That is r=0), I(TT)(TH(HT)I X<IThat is T=1) I(TT)(TH(HT(HH) X≤2( That is z=2) we can see that X-(oo, d)E Fx for all a E R and thus X( is a random variables with respect to Fx A random variable X relative to F maps S into a subset of the real line, and the borel field B on R plays now the role of F. In order to complete the model we need to assign probabilities to the elements b of B. Common sense suggests that the assignment of the probabilities to the event b E B must be consistent with the probabilities assigned to the corresponding events in F. Formally, we need to define a set function Px(: B-0, 1 such that Px(B)=P(X-1(B)≡P(s:X(s)∈B,s∈S) for all B∈B. For example, in the above example, P2({0})=1/4,P2({1}=1/2,P2({2}=1/4andP2({0}U{1})=3/4How do we decide that some function X(·) : S → R is a random variables relative to a given σ-field F ? From the discussion of the σ-field σ(J) generated by the set J = {Bx : x ∈ R} where Bx = (−∞, x] we know that B = σ(J) and if X(·) is such that X −1 ((−∞, x]) = {s : X(s) ∈ (−∞, x], s ∈ S} ∈ F for all (−∞, x] ∈ B, then X −1 (B) = {s : X(s) ∈ B, s ∈ S} ∈ F for all B ∈ B. In other words, when we want to establish that X is a random variables or define Px(·) we have to look no further than the half-closed interval (−∞, x], and the σ-field σ(J) they generate, whatever the range Rx. Let us use the shorthand notation {X(s) ≤ x} instead of {s : X(s) ∈ (−∞, x], s ∈ S} to the above example, X −1 ((−∞, x]) = {s : X(s) ≤ x} =    ∅ X < 0, {(TT)} X ≤ 0 (That is x = 0), {(TT)(T H)(HT)} X ≤ 1 (That is x = 1), {(TT)(T H)(HT)(HH)} X ≤ 2 (That is x = 2), we can see that X−1 ((−∞, x]) ∈ FX for all x ∈ R and thus X(·) is a random variables with respect to FX. A random variable X relative to F maps S into a subset of the real line, and the Borel field B on R plays now the role of F. In order to complete the model we need to assign probabilities to the elements B of B. Common sense suggests that the assignment of the probabilities to the event B ∈ B must be consistent with the probabilities assigned to the corresponding events in F. Formally, we need to define a set function PX(·) : B → [0, 1] such that PX (B) = P(X−1 (B)) ≡ P(s : X(s) ∈ B, s ∈ S) for all B ∈ B. For example, in the above example, Px({0}) = 1/4, Px({1}) = 1/2, Px({2}) = 1/4 and Px({0} ∪ {1}) = 3/4. 15