9/6. Grades. Exam. Card Game. Homework/quizzes: 15% (two lowest scores dropped) Midterms: 25% each Final Exam: 35%

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1 9/6 Wednesday, September 06, 2017 Grades Homework/quizzes: 15% (two lowest scores dropped) Midterms: 25% each Final Exam: 35% Exam Midterm exam 1 Wednesday, October 18 8:50AM - 9:40AM Midterm exam 2 Final exam No Calculator Card Game Wednesday, November 29 8:50AM - 9:40AM Friday, December 15 2:45PM - 4:45PM Simplified Version 1st 2nd 3rd Probability Never Change Cutoff 2 or /6 W W /6 W W /6 L W /6 W W /6 L L /6 L L Probability: 1/2 Probability: 2/3

2 9/8 Friday, September 08, 2017 Probability Space A probability model has three ingredients Sample space The sample space is the set of all possible outcomes Set of all events A subset of We let Probability measure is calle event be the set of all events A probability measure is a function from to that satisfies, for any event Note: and being pairwise disjoint is denonted Equally Likely Outcome Definition When all outcomes are equally likely Example: 431 card game (with full standard deck) Example: 431 card game (with 3 deck) Winning event for the never-change strategy

3 9/11 Monday, September 11, 2017 Probability Space sample space (list of all outcomes) : collection of events (subsets of ) probability measure Equally Likely Outcome Example: 431 game with full deck Example: 431 game with replacement Different Types of Random Experiments Sampling with replacement where order matters Sampling without replacement where order matters

4 Sampling without replacement where order is irrelevant

5 9/13 Wednesday, September 13, 2017 Probability Space Sample Space Collection of events Equally Likely Outcome For finite set with Sampling with replacement where order matters Sampling without replacement where order matters Sampling without replacement where order is irrelevant Note: is called choose Example 1.15

6 Number the shirts Striped: 1, 2, 3, 4, 5 Plaid: 6, 7, 8 Solid: 9, 10 Infinite Sample Spaces Example 1.16

7 Example 1.18

8 9/15 Friday, September 15, 2017 Rule 1 Theorem If are pairwise disjoint Example 1.20

9 Rule 2 Theorem Proof Example 1.21 are disjoint Rule 3 Theorem

10 Rule 3 Theorem Proof Example 1.25 General Inclusion-Exclusion Formula

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12 9/18 Monday, September 18, 2017 Random Variable Definition A random variable is a function from to the real numbers The probability distribution of For all subsets Discrete Random Variable in the collection of probability A discrete random variable takes values in some countable set Probability Mass Function (PMF) Probability mass function is a function Example 1.30 Setup Random Variable Probability mass function of

13 Calculate Let Conditional Probability and Independence Definition be events and assume that Then the conditional probability of given is As a function of, Special case: equally like outcome satisfies the axioms of probability measure Example: 431 game with full deck (without replacement, order matters)

14 9/20 Wednesday, September 20, 2017 Conditional Probability Definition Special case: equally likely Example 2.5 Multiplication Rule Illumination Fact 2.6 If Example 2.7 are events and all the conditional probabilities below make sense, then

15 Partition Definition A partition of is a collection of events that are pairwise disjoint and such that their union is Theorem Let be a partition of Proof For pairwise disjoint evens By multiplication rule Example: 431 game with 4 distinct cards Setup

16 For Therefore when, choose a new card

17 9/22 Friday, September 22, 2017 Conditional Probability Definition More generally For partition of : Example: 431 game with 4 cards Setup For For For For The overall probability of winning

18 Bayes' Rule Definition Let be an event of positive probability Proof Bayes' Rule and Law of Total Probability Example 2.14

19 Independent of Events and are independent if

20 9/25 Monday, September 25, 2017 Independence Definition We says that events and are independent if In particular, if Similarly, if Example 2.19 Independence and Complement Fact 2.20

21 Suppose that and are independent. Then the same is true for each of these pairs Proof Independence of Random Variables Definition Let be random variables defined on the same probability space. Then are independent if For all choices of subsets Discrete Case Discrete random variables of the real line. are independent if and only if For all choices Proof of possible values of the random variables

22 Bernoulli Distribution Binomial Distribution

23 9/27 Wednesday, September 27, :00 AM Independence Independence of Events Events where RVs are independent if for every collection and Independence of Random Variables b Bernoulli Distribution Bernoulli Random Variables Example 4 independent random variables More Generally Binomial Distribution Success in Definition independent trials b Binomial distribution with success probability and trials is denoted as

24 Geometric Distribution Motivation Definition RV Definition number of trials needed to see first success has the geometric distribution with success parameter if the possible values of are {1, 2, 3,... } and fi for positive integers Abbreviate this by Conditional independent Example 2.39 Let be events with. We say and are conditionally independent given B if

25 9/29 Friday, September 29, 2017 Birthday Paradox Assume birthday are equally likely among 365 days, and also independent Suppose people in a room How large does need to be for probability that 2 people have the same birthday Let be the probiability of at least one repeatition among people Let b Discrete Random Variables Recall Definition A random variable is a function from sample space into the real numbers. A random variable fi b fi Probability Mass Function so that b Continuous Random Variables Definition A random variable is continuous if there exists a function Probability Density Function The function For any is called the probability density function of

26 Note could be greater than 1, but it cannot be less than b ε ε ε Example: Uniform Density over Example 3.6 Example