3770 sheet - Laws of Operation: 1. Complement - A A A A U A...

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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. DeMorgans - 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 1s 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 cant 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 *n-k=number of failures *p=prob of successes *q=prob of failures Poisson Distributions: X ~ Pois() ! ) Pr( k k e k X - = = Probability Density Function: X is a continuous RV, f(x) is the pdf if- = 1 ) ( dx x f (area under f(x) is 1)- x x f 2200 , ) ( (always non-negative)- If , A then = A dx x f A X ) ( ) Pr( (prob that X is in a certain region A) = < < 2...
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3770 sheet - Laws of Operation: 1. Complement - A A A A U A...

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