summary of stt - Summary of Stat 581/582 Text A Probability...

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Summary of Stat 581/582 Text: A Probability Path by Sidney Resnick Taught by Dr. Peter Olofsson
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Chapter 1: Sets and Events 2 . 1 : Basic Set Theory SET THEORY The sample space , , is the set of all possible outcomes of an experiment. e.g. if you roll a die, the possible outcomes are ={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. 1-1 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. = 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, Note: A set can be dense and be countable OR uncountable. The power set of is the class of all subsets of , denoted 2 (or P[ ]). (i.e. if has “n” elements, P[ ] has 2 n elements.) Note: if is infinite and countable, then P[ ] is uncountable. SET OPERATIONS Complement: { } Α = Α ϖ ϖ : C Union: { } both or or Β Α = Β Α ϖ ϖ ϖ : e Intersection: { } Β Α = 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.
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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 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 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. 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 σ -field or σ -algebra if: 1. β 2. β β Β Β c 3. β Β Β Β = 1 2 1 ,... , i i A σ -field is closed under complements, countable unions, and intersections. Note: { } , φ is the smallest possible σ -field and 2 is the largest possible σ -field. The sets in a σ -field are called measurable and the space ( 29 β , is called a measurable space .
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6 . 1 : The σ -field Generated by a Given Class C Corollary : The intersection of σ -fields is a σ -field. Let C be a collection of subsets of . The σ -field generated by C , denoted σ( C 29 , is a σ -field satisfying: a) σ( C 29 C b) If B’ is some other σ -field containing C, then B’ σ( C 29.
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