Definition 7 A random variable X is a real valued function from s to R which satisfies the condition that for each Borel set B E B on R, the set X-(B)=s: X(sE B,s∈S} is an event in J Example Define the function X-the number of heads", then X(HH=2, X(THD) 1, X(HTI)=l, and X(TT))=0. Further we see that X-(2)=I(HH)I X-(1)=I(TH), (HT)) and X-(0)=I(TT). In fact, it can be shown that the a-field related to the random variables, X. so defined is 3x={S,0,{(HH)},{(TT)},{(TH),(HT)},{(HH),(TT)} {(HT),(TH),(HH)},{(HT),(TH),(TT)} We can verify that X-(OU(1)=((HT), (TH), ( TT)E Fx, X-GOU {2})={(HH),(TT)}∈ x and X-1({1})u{2})={(HT),(TH),(HH)}∈Fx Ex Consider the random variable Y-number of Head in the first trial". then Y(HHD=Y(HTI=l, and Y(TT)=Y(TH)=0. However Y does not preserve the event structure of Fx since Y-(0=i(TH), (TT)) is not an event in Fx and so does Y-(1)=I(HH), (HT)I From the two examples above, we see that the question"X(: S-+Rx is a random variable "does not make any sense unless some a-field F is also speci- fied. In the case of the function X-number of heads, in the coin-tossing example we see that it is a random variable relative to the a-field Fx. This. however does not preclude y from being a random variable with respect to some other a-field FY; for instance FY=S, 0, (HH), (HT)J, (TH), (TT). Intuition gests that for al value function X(:S-R we should be able define a a-field x on s such that X is a random variable. The concept of a g-field generated by a random variable enables us to concentrate on particular aspects of an experiment without having to consider everything associated with the experiment at the same time. Hence when we choose to define a random variable and the associated a-field we make an implicit choice about the features of the random experiment we are interested inDefinition 7: A random variable X is a real valued function from S to R which satisfies the condition that for each Borel set B ∈ B on R, the set X −1 (B) = {s : X(s) ∈ B, s ∈ S} is an event in F. Example: Define the function X—”the number of heads”, then X({HH}) = 2, X({T H}) = 1, X({HT}) = 1, and X({TT}) = 0. Further we see that X −1 (2) = {(HH)}, X−1 (1) = {(T H),(HT)} and X−1 (0) = {(TT)}. In fact, it can be shown that the σ-field related to the random variables, X, so defined is FX = {S, ∅, {(HH)}, {(TT)}, {(T H),(HT)}, {(HH),(TT)}, {(HT),(T H),(HH)}, {(HT),(T H),(TT)}}. We can verify that X−1 ({0}) ∪ {1}) = {(HT),(T H),(TT)} ∈ FX, X−1 ({0}) ∪ {2}) = {(HH),(TT)} ∈ FX and X−1 ({1})∪{2}) = {(HT),(T H),(HH)} ∈ FX. Example: Consider the random variable Y —”number of Head in the first trial”, then Y ({HH}) = Y ({HT}) = 1, and Y ({TT}) = Y ({T H}) = 0. However Y does not preserve the event structure of FX since Y −1 ({0}) = {(T H),(TT)} is not an event in FX and so does Y −1 ({1}) = {(HH),(HT)} From the two examples above, we see that the question ”X(·) : S → RX is a random variable ?” does not make any sense unless some σ-field F is also speci- fied. In the case of the function X–number of heads, in the coin-tossing example we see that it is a random variable relative to the σ-field FX. This, however, does not preclude Y from being a random variable with respect to some other σ-field FY ; for instance FY = {S, ∅, {(HH),(HT)}, {(T H),(TT)}}. Intuition suggests that for any real value function X(·) : S → R we should be able to define a σ-field FX on S such that X is a random variable. The concept of a σ-field generated by a random variable enables us to concentrate on particular aspects of an experiment without having to consider everything associated with the experiment at the same time. Hence, when we choose to define a random variable and the associated σ-field we make an implicit choice about the features of the random experiment we are interested in. 14