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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 , , 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. 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. = 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: { } = : C Union: { } both or or = : e Intersection: { } = and : SET LAWS Associativity: ( 29 ( 29 C C b b b b = Distributivity: ( 29 T t t T t t = De Morgans: ( 29 T t c t c T t t = * The opposite is true for all of the above. 3 . 1 : Limits of Sets = = n k k k n k inf = = n k k k n k sup = = = = = 1 1 inf inf lim n n k k n k n n n k ={ n : for all but finitely many n} o {A n occurs eventually} = = = = = 1 1 sup sup lim n n k k k n n n n k ={ n : for infinitely many n} o {A n occurs infinitely often} n n n n sup lim inf lim ( 29 c n n c n n = 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. A c A 3. , A A A field is closed under complements, finite unions, and intersections. is called a field or algebra if: 1. 2. c 3. = 1 2 1 ,... , i i A field is closed under complements, countable unions, and intersections. Note: { } , is the smallest possible field and 2 is the largest possible field....
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 Spring '11
 bcd

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