Finite Probability Spaces

Finite Probability SpacesSample space Ω : all possible outcomes of some random experimentExample: In 3-step binomial modelΩ= HHH , HHT , HTH, HTT, THH , THT, TTH,TTT{ }Event : a subset of ΩA = HHH, HHT, HTH,HTTExample: { }HF: Collection of subsets of Ω ( Collection of events of Ω )A = HHH, HHT, HTH , HTTExample: LetH { }AT= { THH, THT, TTH,TTT}F = { ∅, A , A , Ω }HTDefinition 1.1 (probability measure) A probability measure P is a functioninto [0,1] with the following properties:(1). P( Ω ) = 1nn(2). Let A1, A2, L, Anbe disjoint sets in F, then P( U Ai) = ∑ P(Ai)i=1i=1f mapping FExample: F = { ∅, A , A , Ω }HT1=3PAT32=3PAT32(1). PA ( ) , ( ) = is a probability measure.H2(2). PA ( ) , ( ) = is not a probability measure.HDefinition 1.2 (σ -algebra) Let Ω be a nonempty set (sample space). We say acollection of subsets of Ω , say G, is a σ -algebra if(1). ∅∈Gc(2). If A∈G, then A ∈ G(3). Let A1, A2, L, Anbe elements in G, thenExample:1. Ω = { 1,2,3,4,5,6 }G = ∅,{1,2,3},{4,5,6},ΩG is a σ -algebra.{ }nUi=1A ∈G2. { HHH , HHT , HTH, HTT , THH , THT, TTH,TTT}Ω= Leti

Definition 1.10 (measure, Lebesgue measure) LetExampleDefinition 1.11 (Borel-measurable)We say rv. X is Borel-measurable if σ ( X ) is a subset of Borel set.Definition 1.12 (Lebesgue integral)A measure that equal to the length of the interval.The Lebesgue integral has two important advantages over the Riemann integral.The first is that the Lebesgue integral is defined for more functions.The second is that there are three convergence theorems satisfied by the Lebesgueintegral. But there are no such theorems for the Riemann integral,ExamplePlease see ShreveTheorem 3.1 (Fatou’s Lemma)Theorem 3.2 (Monotone Convergence Theorem)Theorem 3.3 (Dominated Convergence Theorem)