extbf {Last Time} extbf {Probability space} extbf {Specifying the Probability Measure $IP $} extbf {Some properties of $IP $} extbf {Definition of Random Variable} extbf {Definition of Random Vector} extbf {Simple Functions} extbf {Approximation with Simple Functions} extbf {Approximation with Simple Functions - illustration} extbf {Closure Properties of RV's} extbf {$sigma $-field generated by a RV} extbf {Example: Exercise 1.2.9} extbf {$sigma $-fields as Information} extbf {Effects of Functions on Information}Last Time•Introduction•Measurable space•Generated σ-fields•Borel σ-fieldToday’s lecture: Sections 1.1–1.2.2MATH136/STAT219 Lecture 2, September 24, 2008 – p. 1/14Probability space•A probability space is a triple (Ω, F, IP ), where (Ω, F) is ameasurable space and IP is a probability measure•A probability measure is a set function IP : F → [0, 1]which satisfies:◦IP (Ω) = 1◦0 ≤ IP (A) ≤ 1 for all A ∈ F◦Countable additivity: if Ai∈ F, i = 1, 2, . . . are mutuallydisjoint (i.e. Ai∩ Aj= ∅, i 6= j) thenIP (∞[i=1Ai) =∞Xi=1IP (Ai)•If IP (A) = 1 then we say A occurs almost surely (a.s.)MATH136/STAT219 Lecture 2, September 24, 2008 – p. 2/14Specifying the Probability Measure IP•Countable Ω◦Set F = 2Ω◦Define pωfor each ω ∈ Ω, such that 0 ≤ pω≤ 1 andPw∈Ωpw= 1◦Then IP (A).=Pω∈Apwdefines a probability measure on(Ω, 2Ω)•Uncountable Ω◦Consider a set of generators {Aα: α ∈ Γ} withF = σ({Aα: α ∈ Γ})◦Define a probability measure IP (Aα) for all Aα, α ∈ Γ◦Then (under mild conditions) IP extends uniquely to aprobability measure on (Ω, F) (see Rosenthal, Section2.3)MATH136/STAT219 Lecture 2, September 24, 2008 – p. 3/14Some properties of IPLet (Ω, F, IP ) be a probability space and A, B, Ai, Bi∈ F,i = 1, 2, . . .•IP (Ac) = 1 − IP (A)•IP (A ∪ B) = IP (A) + IP (B) − IP (A ∩ B)•Monotonicity: if A ⊂ B then IP (A) ≤ IP (B)•Countable subadditivity: if A ⊂ ∪∞i=1AithenIP (A) ≤P∞i=1IP (Ai)•Continuity from below: if Ai↑ A (A1⊂ A2⊂ · · · and∪∞i=1Ai= A) then IP (Ai) ↑ IP (A)•Continuity from above: if Bi↓ B (B1⊃ B2⊃ · · · and∩∞i=1Bi= B) then IP (Bi) ↓ IP (B)MATH136/STAT219 Lecture 2, September 24, 2008 – p. 4/14Definition of Random Variable•A random variable is a real-valued F-measurable functionon (Ω, F)•That is, X : Ω → IR satisfiesX−1(B).= {ω : X(ω) ∈ B} ∈ F, for all B ∈ BEquivalently,X−1((−∞, α]).= {ω : X(ω) ≤ α} ∈ F, for all α ∈ IR•Special case: (Ω, F) = (IRn, Bn). A real-valuedBn-measurable function on (IRn, Bn) is calledBorel-measurable (or a Borel function)•Notation: often write {ω : X(ω) ∈ B} as {X ∈ B}MATH136/STAT219 Lecture 2, September 24, 2008 – p. 5/14Definition of Random Vector•A random vector is an IRn-valued F-measurable functionon (Ω, F)•That is, X = (X1, . . . , Xn) : Ω → IRnsatisfiesX−1(B).= {ω : X(ω) ∈ B} ∈ F, for all B ∈ BnEquivalently, for all αi∈ IR, i = 1, . . . , n,{ω : X1(ω) ≤ α1, . . . , Xn(ω) ≤ αn} ∈ F•Note: X = (X1, . . . , Xn) is a random vector if and only if Xiis a random variable for each i = 1, . . . , nMATH136/STAT219 Lecture 2, September 24, 2008 – p. 6/14Simple Functions•Indicator function (RV) of a set:IA(ω) =(1, ω ∈ A0, ω /∈ A•Simple function (RV):nXi=1ciIAi(w),where c1, . . . , cn∈ IR.Note: can take {Ai} to be mutually disjointMATH136/STAT219 Lecture 2, September 24, 2008 – p. 7/14Approximation with Simple Functions•For any RV X there exists a sequence of simple RV’sXn, n = 1, 2, . . . such that Xn(ω) → X(ω) for all ω ∈ Ω•Step 1: definefn(x) = nI{x>n}+n2n−1Xk=0k2−nI(k2−n,(k+1)2−n](x)•Step 2: if X ≥ 0, setXn(ω) = fn(X(ω))•Step 3: in general, write X = X+− X−and setXn= fn(X+) − fn(X−)MATH136/STAT219 Lecture 2, September 24, 2008 – p. 8/14Approximation with Simple Functions - illustration0 0.5 101234ωX(ω)n=10 0.5 101234ωX(ω)n=20 0.5 101234ωX(ω)n=30 0.5 101234ωX(ω)n=4MATH136/STAT219 Lecture 2, September 24, 2008 – p. 9/14Closure Properties of RV’sLet (Ω, F) be a measurable space and let X1, X2, . . . be asequence of RV’s on it.•If limn→∞Xn(ω) exists and is finite for all w ∈ Ω thenlimn→∞Xnis a RV•If g : IRn→ IR is Borel-measurable, then g(X1, . . . , Xn) is arandom variable•Special cases: the following are random variables◦|X|◦Pni=1αiXi, αi∈ IR◦Qni=1Xi◦max(X1, . . . , Xn) and min(X1, . . . , Xn)◦X+.= max(X, 0) and X−.= − min(X, 0)MATH136/STAT219 Lecture 2, September 24, 2008 – p. 10/14σ-field generated by a RV•The σ-field generated by a RV X, denoted σ(X), is thesmallest σ-field G (⊂ F) for which X is G-measurable•Can show thatσ(X) = σ({X ≤ α}α∈IR)= σ({X ∈ B}B∈B)•If X1, . . . , Xnare random variables on (Ω, F) thenσ(Xi, i = 1, . . . , n) is the smallest σ-field containing σ(Xi)for all i = 1, . . . , nMATH136/STAT219 Lecture 2, September 24, 2008 – p. 11/14Example: Exercise 1.2.9•Consider a sequence of two coin tosses,Ω = {HH, HT, T H, T T }, F = 2Ω•X0= 4•X1= 2X0I{ω1=H}+ 0.5X0I{w1=T }•X2= 2X1I{ω2=H}+ 0.5X1I{w2=T }•Find σ(X0), σ(X1), σ(X2)MATH136/STAT219 Lecture 2, September 24, 2008 – p. 12/14σ-fields as Information•σ(X) contains the events A for which we can saywhether ω ∈ A or not, based solely on the value ofX(ω)•A RV X is G-measurable if and only if the information in G issufficient to determine the value of X.•A RV Y is σ(X1, . . . , Xn)-measurable if and only ifY = g(X1, . . . , Xn) for some Borel-measurable function gMATH136/STAT219 Lecture 2, September 24, 2008 – p. 13/14Effects of Functions on Information•If X1, . . . , Xnare RV’s and g is Borel-measurable, thenσ(g(X1, . . . , Xn)) ⊆ σ(X1, . . . , Xn)•If X1, . . . , Xnand Y1, . . . , Ymare RV’s defined on (Ω, F)such that◦Yk= gk(X1, . . . , Xn) for each k = 1, . . . , m and someBorel-measurable functions gk, and◦Xi= hi(Y1, . . . , Ym) for each i = 1, . . . , n and someBorel-measurable functions hi,thenσ(X1, . . . , Xn) = σ(Y1, . . . , Ym)MATH136/STAT219 Lecture 2, September 24, 2008 – p.