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Lecture 1: Introduction, Set Theory, and Boolean Algebra Syllabus & Policies General Info Classroom: Perkins 2-072 Lecture 1: Introduction, Set Theory, and Boolean Time: Mon - Fri, 2:00 - 3:15 pm Algebra Wed, 3:30 - 4:30 pm Professor: Colin Rundel Sta 111 Office: Old Chemistry 223E Email: [email protected] Colin Rundel Office hours: Monday 12:30 - 1:30 pm May 13, 2014 Monday 3:30 - 5:00 pm or by appointment. Course http:// stat.duke.edu/ ∼cr173/ Sta111 Su14 Website: Sta 111 (Colin Rundel) Lec 1 May 13, 2014 1 / 25 Syllabus & Policies Syllabus & Policies Materials Grading Homework 25% Labs 15% Textbook: Probability and Statistical Inference, Midterm 30% Hogg, Tanis, & Zimmerman Final 30% Pearson, 9th Edition, 2015 ISBN: 9780321923271 Grades will be curved at the end of the course after overall averages have been calculated. Exams: Midterm: Wednesday, June 4th, 2:00 - 4:30 pm Average of 90-100 guaranteed A-. Final: Thursday, June 19t h 2:00 - 4:30 pm Average of 80-90 guaranteed B-. Average of 70-80 guaranteed C-. Holidays: Monday, May 26th - Memorial Day The more evidence there is that the class has mastered the material, the more generous the curve will be. Sta 111 (Colin Rundel) Lec 1 May 13, 2014 2 / 25 Sta 111 (Colin Rundel) Lec 1 May 13, 2014 3 / 25 Syllabus & Policies Syllabus & Policies Labs Policies Objective: Give you hands on experience with data analysis using There will not be make-ups for any of the homework, labs, or exams. statistical software. All regrade requests on homework assignments and exams should be http:// beta.rstudio.org submitted promptly. There will be no grade changes after the final exam. A gmail account is needed to register your Rstudio account, add your gmail address using the quiz on Sakai. Academic Integrity & Duke Community Standard 10 labs are planned. Excused absences Write ups due the following Tuesday - most can be completed in class. Late work policy - for homework and lab write ups: late but during class: -10% after class on due date: -20% next day or later: no credit Sta 111 (Colin Rundel) Lec 1 May 13, 2014 4 / 25 Sta 111 (Colin Rundel) Lec 1 May 13, 2014 5 / 25 Set Theory (Briefly) Set Theory (Briefly) What is a set? Set notation and operations Georg Cantor x 2 A x is an element of A A set is a gathering together into a whole of definite, distinct objects of our perception or of our thought - which are called elements of the set. A [ B, A + B is the union of A and B fxjx 2 A or Bg A \ B, AB is the intersection of A and B fxjx 2 A and Bg A set is a grouping of distinct objects. We will denote sets using capital letters (A,B) and the elements of the set using curly braces (fg). A ⊆ B means A is a subset of B x 2 A ) x 2 B A ⊂ B means A is a proper subset of BA ⊆ B and A 6= B To define a set we either enumerate all elements or use set-builder notation: U is the universal set The set of card suits The set of all prime numbers ;; fg is the empty set f|; }; ~; ♠} fxjx is primeg A0; A; Ac is the complement of AA0 = fxjx 2= Ag jAj is the cardinality of A Sta 111 (Colin Rundel) Lec 1 May 13, 2014 6 / 25 Sta 111 (Colin Rundel) Lec 1 May 13, 2014 7 / 25 Set Theory (Briefly) Set Theory (Briefly) Set operation properties De Morgan's Laws Commutativity A [ B = B [ A Useful rules that allow us to relate conjunction (union / or) and disjuction A \ B = B \ A (intersection / and) using only negation. Associativity (A [ B) [ C = A [ (B [ C) 0 0 0 (A \ B) \ C = A \ (B \ C) (A [ B) = A \ B Distribution A [ (B \ C) = (A [ B) \ (A [ C) 0 0 0 A \ (B [ C) = (A \ B) [ (A \ C) (A \ B) = A [ B Idempotence A [ A = A A \ A = A Absorption A [ (A \ B) = A A \ (A [ B) = A | A [ A0 = U A \ A0 = ; Sta 111 (Colin Rundel) Lec 1 May 13, 2014 8 / 25 Sta 111 (Colin Rundel) Lec 1 May 13, 2014 9 / 25 Set Theory (Briefly) Logic / Boolean Algebra (Briefly) Some examples A little Logic / Boolean Algebra Show the following relations are true: Logic statements are statements that must be either true or false. In general we indicate logic statements using lower case letters (e.g. p,q). 1 A [ B \ C = (A [ B) \ (A [ C) There is a natural correspondence between set theory and logic operators: 2 (A [ C) \ A [ (A \ C) [ C = A [ C Set Theory Logic A [ B p or q 3 B [ A \ (B [ C) [ (B \ C) = B [ (A \ C) A \ B p and q A = B p $ q A ⊆ B p ! q (A [ B)0 = A0 \ B0 (p or q)0 $ p0 and q0 (A \ B)0 = A0 [ B0 (p and q)0 $ p0 or q0 Sta 111 (Colin Rundel) Lec 1 May 13, 2014 10 / 25 Sta 111 (Colin Rundel) Lec 1 May 13, 2014 11 / 25 Logic / Boolean Algebra (Briefly) Introduction to Probability More Correspondence What does it mean to say that: Additionally, we can construct logic statements from sets by asking if an The probability of rolling snake eyes is P(S) = 1=36? element belongs to a set, e.g. The probability of flipping a coin and getting heads is P(H) = 1=2? The probability Apple's stock price goes up today is P(+) = 3=4? A [ B −! x 2 (A [ B) $ x 2 A or x 2 B Interpretations: Symmetry: If there are k equally-likely outcomes, each has Example - let A be the set of possible weather for today, P(E) = 1=k fsun; clouds; rain; snowg: Frequency: If you can repeat an experiment indefinitely, #E P(E) = lim n!1 n Belief: If you are indifferent between winning $1 if E occurs or winning $1 if you draw a blue chip from a box with 100 × p blue chips, rest red, P(E) = p Sta 111 (Colin Rundel) Lec 1 May 13, 2014 12 / 25 Sta 111 (Colin Rundel) Lec 1 May 13, 2014 13 / 25 Introduction to Probability Introduction to Probability Terminology Event space is big Outcome space (Ω) - set of all possible outcomes (!). Even for relatively small outcome spaces there are a lot of possible events Examples: 3 coin tosses fHHH, HHT, HTH, HTT, THH, THT, TTH, TTTg we can define. One die roll f1,2,3,4,5,6g Sum of two rolls f2,3,. ,11,12g Seconds waiting for bus [0; 1) Let Ω = f!1;!2;!3g then Event (E) - subset of Ω (E ⊆ Ω) that might happen, or might not E = f;;! ;! ;! ;! [ ! ;! [ ! ;! [ ! ;! [ ! [ ! ; g Examples: 2 heads fHHT, HTH, THHg 1 2 3 1 2 1 3 2 3 1 2 3 Roll an even number f2,4,6g 3 Wait < 2 minutes [0; 120) jEj = 8 = 2 Random Variable (X ) - a value that depends somehow on chance We can show that for an outcome space Ω = f!1;!2;:::;!ng the total n Examples: # of heads f3, 2, 2, 1, 2, 1, 1, 0g number of possible events is 2 . # flips until heads f3, 2, 1, 1, 0, 0, 0, 0g 2^die f2, 4, 8, 16, 32, 64g n E1 = ;; E2 = !1; E3 = !2;:::; En+1 = !n; En+2 = !1 [ !2;:::; E2n = [ !i i=1 Why can we get away with exclusively using unions here? Sta 111 (Colin Rundel) Lec 1 May 13, 2014 14 / 25 Sta 111 (Colin Rundel) Lec 1 May 13, 2014 15 / 25 Introduction to Probability Introduction to Probability Rules of Probability - Kolmogorov's axioms Useful Identities (1) Non-negative: Complement Rule: P(E) ≥ 0 P(not A) = P(Ac ) = 1 − P(A) (2) Addition: P(E [ F ) = P(E) + P(F ) if E \ F = ; Difference Rule: (2)' Countable Addition: P(B and not A) = P(B \ Ac ) = P(B) − P(A) if A ⊆ B 1 1 X P [ Ei = P(Ei ) if Ei Ej = ; for i 6= j i=1 i=1 Inclusion-Exclusion: (3) Total one: P(A [ B) = P(A) + P(B) − P(AB) P(Ω) = 1 Sta 111 (Colin Rundel) Lec 1 May 13, 2014 16 / 25 Sta 111 (Colin Rundel) Lec 1 May 13, 2014 17 / 25 Introduction to Probability Introduction to Probability Generalized Inclusion-Exclusion Equally Likely Outcomes n X X X n+1 #(E) 1 P( [ Ei ) = P(Ei ) − P(Ei Ej ) + P(Ei Ej Ek ) − ::: + (−1) P(E1 ::: En) X !i 2E i=1 P(E) = = i≤n i<j≤n i<j<k≤n #(Ω) #(Ω) i Notation: For the case of n = 3: Cardinality - #(S) = number of elements in set S P(A [ B [ C) = P(A) + P(B) + P(C) − P(AB) − P(AC) − P(BC) + P(ABC) ( 1 if x 2 S Indicator function - 1x2S = 0 if x 2= S Probability of rolling an even number with a six sided die? E = f2; 4; 6g and Ω = f1; 2; 3; 4; 5; 6g P(E) = 3=6 = 1=2 Sta 111 (Colin Rundel) Lec 1 May 13, 2014 18 / 25 Sta 111 (Colin Rundel) Lec 1 May 13, 2014 19 / 25 Introduction to Probability Introduction to Probability Roulette Roulette, cont. Sta 111 (Colin Rundel) Lec 1 May 13, 2014 20 / 25 Sta 111 (Colin Rundel) Lec 1 May 13, 2014 21 / 25 Sampling Sampling Sampling Birthday Problem Imagine an urn filled with white and black marbles Ignoring leap years, and assuming birthdays are equally likely to be any day .
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