CHAPTER 3: Random Variables and ProbabilityDistributionsReadings: Chapter 3 of the textbook3.1: Random Variables and Distribution FunctionsReview:Recall that B; Borel …eld (or -algebra), is a collection satisfying the following conditions:(i)  2 B;(ii) If A 2 B then Ac2 B;(iii) If Aj2 B; j = 1; 2; :::; then [1j=1Aj2 B:Also, a probability function is a mapping from B to R satisfying the following conditions(i). P (A)  0 for all A 2 B;(ii). P (S) = 1;(iii). If Aj2 B; j = 1; 2; :::; are pairwise mutually exclusive, then P ([1j=1Aj) =P1j=1P (Aj):Suppose P is a probability measure de…ned on the measurable space (S; B). Then (S,B,P) is aprobability space.Example: A Coin is thrown. Then the sample space S = fH; T g:De…neX(H) = 1;X(T ) = 0:Example: Bush Running for the next term of President. S = fWin, Fail}.De…ne X : S ! R byX(Win) = 1;X(Fail) = 0:Example: Three coins are thrown. Then S = fHHH; HHT; HT H; HT T;T HH; T HT; T T H; T T Tg.Event A = ftwo heads appearg = fHHT; HT H; T HHg = 3=8:Example: Throwing a die. S = f1; 2; 3; 4; 5; 6g.Remark: It is inconvenient to work with di¤erent sample spaces. To develop a uni…ed probabilitytheory, we need to unify di¤erent sample spaces. This can be achieved by assigning a real numberto each possible outcome in S. In other words, we construct a mapping from the original samplespace S to a new sample space , a set of real numbers. This transformation X : S ! R iscalled a random variable.1De…nition [Random Variable] A Random variable X(), is a mapping from the sample spaceS to the real line R such that to each outcome s 2 S there exists a corresponding unique realnumber, X(s): The collection of real numbers, X(s); will constitute a new sample space :Remark: The random variable X is a function from S to  :X : S ! :Example: S = fH; T g. De…ne X(H) = 1 and X(T ) = 0: Then  = f1; 0g:Example: S = fWin, Fail}: De…ne X(W in) = 1 and X(F ail) = 0: Then  = f1; 0g:Remark: It is not necessary to have the same number of basic outcomes for S and :Example: S = fT T T; T T H; T HT; HT T; HHT; HT H; T HH; HHHg. De…neX(T; T; T ) = 0;X(T; T; H) = 1;X(T; H; T ) = 1; X(H; T; T ) = 1; X(H; H ; T ) = 2; X(H; T; H) = 2; X(T; H; H) = 2;X(H; H; H) = 3: Then  = f0; 1; 2; 3g:P (X = 2) =?Intuition: X(s) is the number of heads. Therefore, P(X = 3) = P (A); where A = fs inS : X(s) = 3g; denotes the probability that exactly three heads o ccur in the experiment.Example: S = f1; 2; 3; 4; 5; 6g: De…ne X(s) = s: Then  = S: Identity transformation. Neednot change S because S consists of numbers already.Remark: The transformation need not be 1-1.Question: Suppose the number of basic outcomes in S is countable. (a) Is it possible that thenumber of basic outcomes in  is larger than that of S? (b) Smaller than that of S?Example: S = fs : 1 < s < 1g: X(s) = 1 if s > 0 and X(s) = 0 if s  0:Remark: This is useful for directional forecasts or investigation of asymmetric business cycles(e.g., Neftci 1984, "Are Economic Time Series Asymmetric Over the Business Cycles?", JPE 92,307-328).Remark: To assign the probability measure to X, such as “the probability that X lies betweena and b is 0.5”, we must de…ne probabilities for random variables. This is achieved by relatingprobability statements on X to probability statements about the corresponding subsets of S:Suppose we are interested in P (a < X < b):P : B ! R+;B  S:2Step 1: De…ne an event in SA = fs 2 S : a < X(s) < bg= collection of all basic outcomes for which X(s) 2 (a; b):That is, A is a collection of all possible basic outcomes s in S that satisfy the condition ofa < X(s) < b:Step 2: ThenP [a < X < b] = P (A) =Xs2APr(s):Example (Three Coins thrown): S = fHHH; :::; T T T g;  = f0; 1; 2; 3g:P [0  X  1] = P fs 2 S : 0  X(s)  1g:A = fs 2 S : 0  X(s)  1g= fT T T; T T H; T HT; HT T g:P [0  X  1] = P(A)= 1=8 + 1=8 + 1=8 + 1=8=12:However, when S is continuous (e.g. S = R); we cannot use the above expressions immediatelyunless we can be sure that the set A = fs 2 S : 0  X(s)  1g belongs to the -algebra B.Whether or not A 2 B depends on, of course, on the form of mapping X().What functional form X() will ensure A 2 B? Or what condition on X() will ensure thatthe set A is in B?The following condition ensures that A always belongs to B:De…nition [Measurable Function]: A function X on S to R is B-measurable (or measurablewith respect to the sigma …eld B) if for every real number a the set [s 2 S : X(s)  a] 2 B:Remarks:(i) When a function is B-measurable, we can express the probability of an event, say \X a00; in terms of the probability of another event in B; say, {s 2 S : X(s)  ag: In otherwords, the measurable function ensures that P (X 2 ) is always well-de…ned. If X() is nota measurable function, then there exist subsets on R for which probabilities are not de…ned.However, constructing such sets are very complicated.(b) In what follows, we assume that the random variable is always measurable with respectto some -algebra of S. In fact, in the advanced probability theory, the term “random variable”is restricted to a B-measurable function from S ! R:3De…nition [Realization]: Suppose a random experiment gives rise to some number x for R.V.X: We will call x is a realization of R.V. X:Question: How does a realization x come up? A random experiment gives rise to an outcomes; which gives the realization x = X(s):Convention: The capital letter X denotes the random variable, the lowercase letter x denotesa possible value that X can take.Review:(i) (S; B; P );(ii) X : S ! ;(iii) (; BX; PX); where PXis called an induced probability function, because it can be derivedfrom the orginal probability function:PX(A) = P (X 2 A)= P (C);whereC = fs 2 S : X(s) 2 Ag  B:An example of A isA = fx 2  : a < X < bg:Question: How to characterize a random variable X?We can of course use the induced probability function PX(): However, there exists a moreconvenient way to characterize the distribution of X: This is called the cumulative distributionfunction, to be de…ned below.X; x1; x2; :::; xn; P (X = xi) =?X = direction of IBM price change, = f1; 1; 0gP (X = …