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Unformatted text preview: Summary of Stat 581/582 Text: A Probability Path by Sidney Resnick Taught by Dr. Peter Olofsson Chapter 1: Sets and Events ∫ 2 . 1 : Basic Set Theory SET THEORY The sample space , Ω , is the set of all possible outcomes of an experiment. e.g. if you roll a die, the possible outcomes are Ω ={1,2,3,4,5,6}. A set is finite / infinitely countable if it has finitely many points. A set is countable / denumerable if there exists a bijection (i.e. 11 mapping) to the natural numbers, N ={1,2,3,…} e.g. odd numbers, integers, rational numbers (Q = {m/n} where m and are integers) The opposite of countable is uncountable . e.g. ℜ = real numbers, any interval (a,b) If you take any two points in a set and there are infinitely many points between them, then the set is considered to be dense . e.g. Q, ℜ Note: A set can be dense and be countable OR uncountable. The power set of Ω is the class of all subsets of Ω , denoted 2 Ω (or P[ Ω ]). (i.e. if Ω has “n” elements, P[ Ω ] has 2 n elements.) Note: if Ω is infinite and countable, then P[ Ω ] is uncountable. SET OPERATIONS • Complement: { } Α ∉ Ω ∈ = Α ϖ ϖ : C • Union: { } both or or Β ∈ Α ∈ Ω ∈ = Β Α ϖ ϖ ϖ : e • Intersection: { } Β ∈ Α ∈ Ω ∈ = Β Α ϖ ϖ ϖ and : SET LAWS • Associativity: ( 29 ( 29 C C b b b b Β Α = Β Α • Distributivity: ( 29 T t t T t t ∈ ∈ Α Β = Α Β • De Morgan’s: ( 29 T t c t c T t t ∈ ∈ Α = Α * The opposite is true for all of the above. ∫ 3 . 1 : Limits of Sets • ∞ = Α = Α ≥ n k k k n k inf • ∞ = Α = Α ≥ n k k k n k sup • ∞ = ∞ = ∞ = Α = Α ≥ = Α 1 1 inf inf lim n n k k n k n n n k ={ n Α ∈ Ω ∈ ϖ ϖ : for all but finitely many n} o {A n occurs eventually} • ∞ = ∞ = ∞ = Α = Α ≥ = Α 1 1 sup sup lim n n k k k n n n n k ={ n Α ∈ Ω ∈ ϖ ϖ : for infinitely many n} o {A n occurs infinitely often} • n n n n Α ⊆ Α sup lim inf lim • ( 29 c n n c n n Α = Α sup lim inf lim (and vice versa) If liminf = limsup = A n , then the limit exists and is A n . ∫ 5 . 1 : Set Operations and Closure A is called a field or algebra if: 1. ∈ Ω A 2. ∈ Α A ∈ Α ⇒ c A 3. ∈ Β Α , A ∈ Β Α ⇒ A A field is closed under complements, finite unions, and intersections. β is called a σfield or σalgebra if: 1. β ∈ Ω 2. β β ∈ Β ⇒ ∈ Β c 3. β ∈ Β ⇒ Β Β ∞ = 1 2 1 ,... , i i A σfield is closed under complements, countable unions, and intersections. Note: { } Ω , φ is the smallest possible σfield and Ω 2 is the largest possible σfield....
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This note was uploaded on 04/27/2011 for the course ACCT 2311 taught by Professor Bcd during the Spring '11 term at University of Houston.
 Spring '11
 bcd

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