3770 sheet

# 3770 sheet - Laws of Operation 1 Complement A A A A U A A =...

This preview shows pages 1–2. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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 *n-k=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 non-negative)- If , ℜ ⊆ A then ∫ = ∈ A dx x f A X ) ( ) Pr( (prob that X is in a certain region A) ∫ = < < 2...
View Full Document

{[ snackBarMessage ]}

### Page1 / 3

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online