STAT 810 Probability Theory I Chapter 3: Random Variables, Elements, and Measurable Maps Dr. Dewei Wang Associate Professor Department of Statistics University of South Carolina [email protected] 1 / 25 Introduction We will precisely define a random variable. A random variable is a real valued function with domain Ω which has an extra property called measurability that allows us to make probability statements about the random variable. 2 / 25 3.1 Inverse Maps 0 0 Suppose Ω and Ω are two sets. Frequently, Ω = R. Suppose X :Ω 7! Ω0 meaning X is a function with domain Ω and range Ω0. Then X determines a function X −1 : P(Ω0) !P(Ω) defined by X −1(A0) = f! 2 Ω: X (!) 2 A0g for A0 ⊂ Ω0. For example, Ω = fhh; ht; th; ttg collects all possible results of flip- ping a coin twice, X denotes the number of heads which is a map from Ω to Ω0 = f0; 1; 2g, where X (hh) = 2, X (ht) = X (th) = 1, and X (tt) = 0. 3 / 25 3.1 Inverse Maps The X −1 preserves complementation, union, and intersections as the 0 0 0 0 following properties show. For A ⊂ Ω , At ⊂ Ω , and T an arbitrary index set, we have (i) X −1(;) = ; and X −1(Ω0) = Ω. (ii) X −1(A0c ) = fX −1(A0)gc or X −1fΩ0 n A0g = Ω n X −1(A0). −1 0 −1 0 (iii) X ([t2T At ) = [t2T X (At ) and −1 0 −1 0 X (\t2T At ) = \t2T X (At ). 4 / 25 3.1 Inverse Maps Notation: If C0 2 P(Ω0) is a class of subsets of Ω0, define X −1(C0) = fX −1(C 0): C 0 2 C0g: Proposition 3.1.1 If B0 is a σ-algebra of subsets of Ω0, then X −1(B0) is a σ-algebra of subsets of Ω. Proposition 3.1.2 If C0 is a class of subsets of Ω0 then X −1(σ(C0)) = σ(X −1(C0)) that is, the inverse image of the σ-algebra generated by C0 in Ω0 is the same as the σ-algebra generated in Ω by the inverse image. 5 / 25 3.1 Inverse Maps Proof of Proposition 3.1.2 From Proposition 3.1.1, X −1(σ(C0)) is a σ-algebra, and since σ(C0) ⊃ C0, X −1(σ(C0)) ⊃ X −1(C0). Therefore X −1(σ(C0)) ⊃ σ(X −1(C0)). Conversely, define F 0 = fB0 2 P(Ω0): X −1(B0) 2 σ(X −1(C0))g ⊃ C0: Then F 0 is a σ-algebra since 1. Ω0 2 F 0, since X −1(Ω0) = Ω 2 σ(X −1(C0)). 2. A0 2 F 0 implies A0c 2 F 0 since X −1(A0c ) = (X −1(A0))c . 0 0 0 0 3. Bn 2 F implies [nBn 2 F since −1 0 −1 0 −1 0 X ([nBn) = [nX (Bn) 2 σ(X (C )). By definition, X −1(F 0) ⊂ σ(X −1(C0)) and C0 ⊂ F 0. Because F 0 is an algebra, σ(C0) ⊂ F 0. Thus X −1(σ(C0)) ⊂ X −1(F 0). Done. 6 / 25 3.2 Measurable Maps, Random Elements, Induced Probability Measures A pair (Ω; B) consisting of a set and a σ-field of subsets is called a measurable space. If (Ω; B) and (Ω0; B0) are two measurable spaces, then a map X :Ω ! Ω0 is called measurable if X −1(B0) ⊂ B: X is also called a random element of Ω0. We will use the notation that X 2 B=B0 or X : (Ω; B) 7! (Ω0; B0): 0 0 A special case occurs when (Ω ; B ) = (R; B(R)). In this case, X is called a random variable. (e.g., X = IA is a random variable iff A 2 B). 7 / 25 3.2 Measurable Maps, Random Elements, Induced Probability Measures Let (Ω; B; P) be a probability space and suppose X : (Ω; B) 7! (Ω0; B0) is measurable. Define for A0 ⊂ Ω0, [X 2 A0] = X −1(A0) = f! : X (!) 2 A0g: Define the set function P ◦ X −1 on B0 by P ◦ X −1(A0) = P(X −1(A0)); P ◦ X −1 is a probability on (Ω0; B0) called the induced probability on the distribution of X . Usually, we write P ◦ X −1(A) = P[X 2 A0]: If X is a random variable, then P ◦ X −1 is the measure induced on −1 R by the distribution function P ◦ X (−∞; x] = P[X ≤ x]. 8 / 25 Example Consider the experiment of tossing two die and let Ω = f(i; j): 1 ≤ i; j ≤ 6g and Define X :Ω 7! f2; 3;:::; 12g = Ω0 by X ((i; j)) = i + j. Then X −1(f4g) = [X 2 f4g] = [X = 4] = f(1; 3); (3; 1); (2; 2)g ⊂ Ω and X −1(f2; 3g) = [X 2 f2; 3g] = f(1; 1); (1; 2); (2; 1)g ⊂ Ω: The distribution of X is the probability measure on Ω0 specified by P ◦ X −1(fig) = P[X = i]; i 2 Ω0: 9 / 25 3.2 Measurable Maps, Random Elements, Induced Probability Measures We now verify P ◦ X −1 is a probability measure on B0: (a) P ◦ X −1(Ω0) = P(Ω) = 1 (b) P ◦ X −1(A0) = P(X −1(A0)) ≥ 0 0 0 −1 0 (c) if fAng are disjoint in B , then fX (An)g are disjoint in B, −1 0 −1 0 −1 0 P ◦ X ([nAn) = P(X ([nAn)) = P([nX (An)) X −1 0 X −1 0 = P(X (An)) = P ◦ X (An): n n Remark: When X is a random element of B0, we can make probability statements about X , since X −1(B0) 2 B and the probability measure P knows how to assign probabilities to elements of B. The concept of measurability is logically necessary in order to be able to assign probabilities to sets determined by random elements. 10 / 25 3.2 Measurable Maps, Random Elements, Induced Probability Measures The definition of measurability makes it seem like we have to check X −1(A0) 2 B for every A0 2 B0; that is X −1(B0) ⊂ B. In fact, it usually suffices to check that X −1 is well behaved on a smaller class than B0. Proposition 3.2.1 (Test for measurability) Suppose X :Ω 7! !0 where (Ω; B) and (Ω0; B0) are two measurable spaces. Suppose C0 generates B0; that B0 = σ(C0): Then X is measurable iff X −1(C0) ⊂ B: 11 / 25 3.2 Measurable Maps, Random Elements, Induced Probability Measures Proof of Proposition 3.2.1. if X −1(C0) ⊂ B, then by minimality σ(X −1(C0)) ⊂ B. However we get X −1(σ(C0)) = X −1(B0) = σ(X −1(C0)) ⊂ B; which is the definition of measurability. Corollary 3.2.1 (Special case of random variables) The real valued function X :Ω 7! R is a random variable iff −1 X ((−∞; x]) = [X ≤ x] 2 B, for x 2 R. Proof of Corollary 3.2.1 This follows directly from σ((−∞; x]; x 2 R) = B(R): 12 / 25 3.2.1 Composition Proposition 3.2.2 (Composition) (HW 3-1: prove this proposition) Let X1, X2 be two measurable maps X1 : (Ω1; B1) 7! (Ω2; B2) and X2 : (Ω2; B2) 7! (Ω3; B3) where (Ωi ; Bi ), i = 1; 2; 3 are measurable spaces. Define X = X2 ◦ X1 :Ω1 7! Ω3 by X (!) = X2 ◦ X1(!1) = X2(X1(!1));!1 2 Ω1. Then X = X2 ◦ X1 2 B1=B3: 13 / 25 3.2.2 Random Elements of Metric Spaces The most common use of the name random elements is when the range is a metric space. Let (S; d) be a metric space with metric d so that d : S × S 7! R+ satisfies (i) d(x; y) ≥ 0. (ii) d(x; y) = 0 iff x = y. (iii) d(x; y) = d(y; x). (iv) d(x; z) ≤ d(x; y) + d(y; z). Let O be the class of open subsets of S. Define the Borel σ-algebra S = σ(O). If X : (Ω; B) 7! (S; S), that is X 2 B=S, then call X a random element of S. 14 / 25 3.2.2 Random Elements of Metric Spaces Noteworthy Examples 1. S = R, d(x; y) = jx − yj, a random element X of S is called a random variable. k 2. S = R , d(xx;xyy)y = kxx−x yyky2 (Euclidean norm), a random element XX=X (X1;:::; Xk ) of S is called a random vector. 1 3. S = R , 1 Pk ! X jxi − yi j d(xx;xyy)y = 2−k i=1 ; Pk k=1 1 + i=1 jxi − yi j a random element XX=X (X1; X2;::: ) of S is called a random sequence. 15 / 25 3.2.2 Random Elements of Metric Spaces Noteworthy Examples (continued) 4. S = C[0; 1) be the set of all real valued continuous functions with domain [0; 1). Define kx − ykm = sup jx(t) − y(t)j 0≤t≤m and 1 X kx − ykm d(x; y) = 2−m ; 1 + kx − yk m=1 m a random element X = X (·) of S is called a random (continuous) function. 16 / 25 3.2.3 Measurability and Continuity Proposition 3.2.3 Suppose (Si ; di ); i = 1; 2 are two metric spaces. Let the Borel σ-algebra (generated by open sets) be Si , i = 1; 2. If X : S1 ! S2 is continuous, then X is measurable: X 2 S1=S2.