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Unformatted text preview: Laws of Operation: 1. Complement  A A A A U A A = ∅ = ∩ = ∪ , , 2. Commutative  A B B A A B B A ∩ = ∩ ∪ = ∪ , 3. DeMorgan’s  B A B A B A B A ∪ = ∩ ∩ = ∪ , 4. Associative  ( 29 ( 29 ( 29 ( 29 C B A C B A C B A C B A ∩ ∩ = ∩ ∩ ∪ ∪ = ∪ ∪ , 5. Distributive  ) ( ) ( ) ( ), ( ) ( ) ( C A B A C B A C A B A C B A ∩ ∪ ∩ = ∪ ∩ ∪ ∩ ∪ = ∩ ∪ A number of elements in A (cardinality) ) Pr( 1 ) Pr( A A = For any 2 events: ) Pr( ) Pr( ) Pr( ) Pr( B A B A B A ∩ + = ∪ For any 3 events: ) Pr( ) Pr( ) Pr( ) Pr( ) Pr( ) Pr( ) Pr( ) Pr( C B A C B C A B A C B A C B A ∩ ∩ + ∩ ∩ ∩ + + = ∪ ∪ Multiplication Rule: For any event A in a SSS S : elementsS elementsA S A A # # ) Pr( = = Permutations: )! ( ! , r n n r n P = n  # of things r  # of things taken at a time *concerned with order ) , , ( ) , , ( c a b c b a ≠ Combinations: r n or r n C , *combinations of n things taken r at a time *not concerned with order ) , , ( ) , , ( c a b c b a = ! )! ( ! r r n n r n = Hypergeometric Distributions: For a objects of type 1 and b objects of type 2: Select n objects w/o replacement from a+b Pr(k type 1’s were picked) +  = n b a k n b k a Conditional Probability: ) Pr( ) Pr( ) Pr( B B A B B A B A ∩ = ∩ = If Pr(B) > 0, then A and B are indep. Iff ) Pr( ) Pr( A B A = o Independence – prob of A does not depend on whether or not B occurs If Pr(A) > 0 and Pr(B) > 0, then A and B can’t be indep and disjoint at the same time. A, B, C are indep iff: ) Pr( ) Pr( ) Pr( ) Pr( C B A C B A = ∩ ∩ All pairs must be indep ) Pr( ) Pr( ) Pr( ) Pr( ) Pr( ) Pr( ) Pr( ) Pr( ) Pr( C B C B C A C A B A B A = ∩ = ∩ = ∩ Bayes Theorem: ∑ = = n j j A B j A i A B i A B i A 1 ) Pr( ) Pr( ) Pr( ) Pr( ) Pr( Binomial Distribution: If X ~ Bin(n,p), then prob of k successes in n trials is: k n q k p k n k X = = ) Pr( *k=number of successes *nk=number of failures *p=prob of successes *q=prob of failures Poisson Distributions: X ~ Pois(λ) ! ) Pr( k k e k X λ λ = = Probability Density Function: X is a continuous RV, f(x) is the pdf if… ∫ ℜ = 1 ) ( dx x f (area under f(x) is 1) x x f 2200 ≥ , ) ( (always nonnegative) If , ℜ ⊆ A then ∫ = ∈ A dx x f A X ) ( ) Pr( (prob that X is in a certain region A) ∫ = < < 2...
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This note was uploaded on 04/14/2008 for the course ISYE 3770 taught by Professor Goldsman during the Fall '07 term at Georgia Tech.
 Fall '07
 goldsman

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