SDS 321: Introduction to Probability and Statistics
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1 SDS 321: Introduction to Probability and Statistics Lecture 1: Axioms of Probability Purnamrita Sarkar Department of Statistics and Data Science The University of Texas at Austin psarkar/teaching 1
2 Getting Started Your instructor: Prof. Purna Sarkar Office Hours: Tuesdays 11:30-12:30, GDC Your TA: TA Office Hours: Krishna Teja Rekapalli Wednesdays 5-7pm 2
3 Course Overview This course provides an introduction to probability and statistics. The first section will be on fundamentals of probability, including: Discrete and continuous random variables Combinatorics Multiple random variables Functions of random variables Limit theorems The second section will be on statistics, including: Parameter estimation Hypothesis testing We will consider mainly classical statistics. If time permits we will discuss Bayesian Statistics. 3
4 Course materials Course syllabus, slides and homework assignments will be posted at psarkar/teaching Grades will be posted at canvas.utexas.edu The course text books are 1. Introduction to Probability, by Dimitri P. Bertsekas and John N. Tsitsiklis. 2. A First Course in Probability, by Sheldon Ross Another good book that covers similar material is Introduction to Probability, by Charles M. Grinstead and Laurie J. Snell 4
5 Assessment 4 exams. 2 midterms, and the final will consist of 2 midterm length exams. I will take the best 3 out of 4. The final grade will be 25% Homework, 25% from all 3 exams. Homework will be assigned (approximately) weekly, with roughly 10 homeworks in total. Homeworks should be submitted via Canvas by 5pm one week after it is assigned. 5
6 What is probability? If I flip this coin, the probability of getting heads is 0.5 What does this mean? 6
7 What is probability? If I flip this coin, the probability of getting heads is 0.5 What does this mean? If I were to toss the coin 10 times, roughly 5 times I will see a head. A probability of 1 means it is certain, a probability of 0 means it is impossible. In general, you do an experiment many times, and you count how many times a particular event occurs. The proportion roughly gives you the probability of that particular event. 6
8 Experiments and events? Experiment: Tossing a coin twice Experiment: You throw two dice 7
9 Experiments and events? Experiment: Tossing a coin twice Event: you get two heads Experiment: You throw two dice 7
10 Experiments and events? Experiment: Tossing a coin twice Event: you get two heads Event: you get two different outcomes Experiment: You throw two dice 7
11 Experiments and events? Experiment: Tossing a coin twice Event: you get two heads Event: you get two different outcomes Experiment: You throw two dice Event: the sum of the rolls is six 7
12 Experiments and events? Experiment: Tossing a coin twice Event: you get two heads Event: you get two different outcomes Experiment: You throw two dice Event: the sum of the rolls is six Event: you get two odd faces. 7
13 Sample space The sample space Ω is the set of all possible outcomes of an experiment. 8
14 Sample space The sample space Ω is the set of all possible outcomes of an experiment. You rolled one die. What is Ω?. 8
15 Sample space The sample space Ω is the set of all possible outcomes of an experiment. You rolled one die. What is Ω? {1, 2, 3, 4, 5, 6}. 8
16 Sample space The sample space Ω is the set of all possible outcomes of an experiment. You rolled one die. What is Ω? {1, 2, 3, 4, 5, 6}. You tossed three coins together. What is Ω?. 8
17 Sample space The sample space Ω is the set of all possible outcomes of an experiment. You rolled one die. What is Ω? {1, 2, 3, 4, 5, 6}. You tossed three coins together. What is Ω? {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT }. The different elements of a sample space must be mutually exclusive and collectively exhaustive. 8
18 Sample space The sample space Ω is the set of all possible outcomes of an experiment. You rolled one die. What is Ω? {1, 2, 3, 4, 5, 6}. You tossed three coins together. What is Ω? {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT }. The different elements of a sample space must be mutually exclusive and collectively exhaustive. Ω for three coin tosses cannot be {at least one head, at most one tail}. 8
19 Sample space The sample space Ω is the set of all possible outcomes of an experiment. You rolled one die. What is Ω? {1, 2, 3, 4, 5, 6}. You tossed three coins together. What is Ω? {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT }. The different elements of a sample space must be mutually exclusive and collectively exhaustive. Ω for three coin tosses cannot be {at least one head, at most one tail}. An event is a collection of possible outcomes. 8
20 Simple and compound events Simple event: Your two coin tosses came up HH. Your rolled die shows a 6 Compound event: can be decomposed into simple events Your two coin tosses give two different outcomes The sum of the two rolled dice is six You got two odd faces from rolling two dice. 9
21 Simple and compound events Simple event: Your two coin tosses came up HH. Your rolled die shows a 6 Compound event: can be decomposed into simple events Your two coin tosses give two different outcomes You got HT or TH. The sum of the two rolled dice is six You got two odd faces from rolling two dice. 9
22 Simple and compound events Simple event: Your two coin tosses came up HH. Your rolled die shows a 6 Compound event: can be decomposed into simple events Your two coin tosses give two different outcomes You got HT or TH. The sum of the two rolled dice is six You got (1, 5) or (2, 4) or (3, 3) or (4, 2) or (5, 1). You got two odd faces from rolling two dice. 9
23 Simple and compound events Simple event: Your two coin tosses came up HH. Your rolled die shows a 6 Compound event: can be decomposed into simple events Your two coin tosses give two different outcomes You got HT or TH. The sum of the two rolled dice is six You got (1, 5) or (2, 4) or (3, 3) or (4, 2) or (5, 1). You got two odd faces from rolling two dice. You got (1, 1), or (1, 3) or.... 9
24 Sets and sample spaces We need to introduce some mathematical concepts to define probability more concretely: A set is a collection of objects, which are called elements The natural numbers are a set, where the elements are individual numbers. This class is the set, where the elements are the professor, the TA and the students. If an element x is in a set S, we write x S. If a set contains no elements, we call it the empty set,. If a set contains every possible element, we call it the universal set, Ω. 10
25 Sets A set can be finite (e.g. the set of people in this class) or infinite (e.g. the set of real numbers). Set of primary colors = {red, blue, yellow}. 11
26 Sets A set can be finite (e.g. the set of people in this class) or infinite (e.g. the set of real numbers). Set of primary colors = {red, blue, yellow}. If we can enumerate the elements of an infinite set, i.e. arrange the elements in a list, we say it is countable. Set of positive integers = {1, 2,... } 11
27 Sets A set can be finite (e.g. the set of people in this class) or infinite (e.g. the set of real numbers). Set of primary colors = {red, blue, yellow}. If we can enumerate the elements of an infinite set, i.e. arrange the elements in a list, we say it is countable. Set of positive integers = {1, 2,... } If we cannot enumerate the elements, we say it is uncountable. the real numbers the set of all subsets of natural numbers, aka the power set 11
28 Sets A set can be finite (e.g. the set of people in this class) or infinite (e.g. the set of real numbers). Set of primary colors = {red, blue, yellow}. If we can enumerate the elements of an infinite set, i.e. arrange the elements in a list, we say it is countable. Set of positive integers = {1, 2,... } If we cannot enumerate the elements, we say it is uncountable. the real numbers the set of all subsets of natural numbers, aka the power set We can use curly brackets to describe a set in terms of its elements: Sample space of a die roll: S = {1, 2, 3, 4, 5, 6} Arbitrary set where all the elements meet some criterion C: S = {x x satisfies C} 11
29 Operations on sets Let the universal set Ω be the set of all objects we might possibly be interested in. The complement, S c, of a set S, w.r.t. Ω, is the set of all elements that are in Ω but not in S. So Ω c =. We say S T, if every element in S is also in T. S T and T S if and only if S = T. Ω Ω S T S S c is the shaded region S T Ω 12
30 Operations on sets: Union, Intersection, Difference The union, S T, of two sets S and T is the set of elements that are in either S or T (or both): S T = {x x S or x T }. 13
31 Operations on sets: Union, Intersection, Difference The union, S T, of two sets S and T is the set of elements that are in either S or T (or both): S T = {x x S or x T }. The intersection, S T, of two sets S and T is the set of elements that are in both S and T : S T = {x x S and x T } 13
32 Operations on sets: Union, Intersection, Difference The union, S T, of two sets S and T is the set of elements that are in either S or T (or both): S T = {x x S or x T }. The intersection, S T, of two sets S and T is the set of elements that are in both S and T : S T = {x x S and x T } The difference, S \ T, of two sets S and T is the set of elements that are in S, but not in T : S \ T = {x x S and x / T } 13
33 Operations on sets: Union, Intersection, Difference The union, S T, of two sets S and T is the set of elements that are in either S or T (or both): S T = {x x S or x T }. The intersection, S T, of two sets S and T is the set of elements that are in both S and T : S T = {x x S and x T } The difference, S \ T, of two sets S and T is the set of elements that are in S, but not in T : S \ T = {x x S and x / T } Ω Ω Ω S T S T S T S T S T S \ T = S T c 13
34 Operations on sets We can extend the notions of union and intersection to multiple (even infinitely many!) sets: n Sn =S 1 S 2 Sn = {x x Sn for some 1 i n} i=1 n Sn =S 1 S 2 Sn = {x x Sn for all 1 i n} i=1 We say two sets are disjoint if their intersection is empty. We say a collection of sets are disjoint if no two sets have any common elements. If a collection of disjoint sets have union S, we call them a partition of S. 14
35 Probability laws The probability law assigns to an event E a non-negative number P(E) which encodes our belief/knowledge about the likelihood of the event E. Axioms of probability: 15
36 Probability laws The probability law assigns to an event E a non-negative number P(E) which encodes our belief/knowledge about the likelihood of the event E. Axioms of probability: Nonnegativity: P(A) 0, for every event A. Additivity: If A and B are two disjoint events, then the probability of their union satisfies P(A B) = P(A) + P(B). This extends to the union of infinitely many disjoint events: P(A 1 A 2... ) = P(A 1) + P(A 2) +... Normalization: The probability of the entire sample space Ω is equal to 1, i.e. P(Ω) = 1 15
37 Examples You tossed two fair dice together. What is the probability of the event E = {sum of the rolls = 6}? 16
38 Examples You tossed two fair dice together. What is the probability of the event E = {sum of the rolls = 6}? The sample space is {(i, j) 1 i, j 6}. There are a total of 36 outcomes. 16
39 Examples You tossed two fair dice together. What is the probability of the event E = {sum of the rolls = 6}? The sample space is {(i, j) 1 i, j 6}. There are a total of 36 outcomes. Since the dice are fair, each outcome is equally likely. This means every outcome has probability 1/36. 16
40 Examples You tossed two fair dice together. What is the probability of the event E = {sum of the rolls = 6}? The sample space is {(i, j) 1 i, j 6}. There are a total of 36 outcomes. Since the dice are fair, each outcome is equally likely. This means every outcome has probability 1/36. First decompose into simple events. We get a sum of 6 if we get (1, 5) or (2, 4) or (3, 3) or (4, 2) or (5, 1). 16
41 Examples You tossed two fair dice together. What is the probability of the event E = {sum of the rolls = 6}? The sample space is {(i, j) 1 i, j 6}. There are a total of 36 outcomes. Since the dice are fair, each outcome is equally likely. This means every outcome has probability 1/36. First decompose into simple events. We get a sum of 6 if we get (1, 5) or (2, 4) or (3, 3) or (4, 2) or (5, 1). Using the additivity law we have P(E) = 5/36. 16
42 Examples You tossed two fair dice together. What is the probability of the event E = {sum of the rolls = 6}? The sample space is {(i, j) 1 i, j 6}. There are a total of 36 outcomes. Since the dice are fair, each outcome is equally likely. This means every outcome has probability 1/36. First decompose into simple events. We get a sum of 6 if we get (1, 5) or (2, 4) or (3, 3) or (4, 2) or (5, 1). Using the additivity law we have P(E) = 5/36. This is an example of an uniform distribution, where all outcomes are equally likely. 16
43 Properties of probability laws All the following can be proven by decomposing a set into disjoint partitions and using the additivity and non-negativity rules. If A B, then P(A) P(B). 17
44 Properties of probability laws All the following can be proven by decomposing a set into disjoint partitions and using the additivity and non-negativity rules. If A B, then P(A) P(B). B = A (A \ B), where A and B \ A are disjoint. So P(B) = P(A) + P(B \ A) P(A). 17
45 Properties of probability laws All the following can be proven by decomposing a set into disjoint partitions and using the additivity and non-negativity rules. If A B, then P(A) P(B). B = A (A \ B), where A and B \ A are disjoint. So P(B) = P(A) + P(B \ A) P(A). P(A B) = P(A) + P(B) P(A B). 17
46 Properties of probability laws All the following can be proven by decomposing a set into disjoint partitions and using the additivity and non-negativity rules. If A B, then P(A) P(B). B = A (A \ B), where A and B \ A are disjoint. So P(B) = P(A) + P(B \ A) P(A). P(A B) = P(A) + P(B) P(A B). A B=(A \ B) B. So the additivity rule gives P(A B) = P(A \ B) + P(B). Can you finish the proof? A \ B A B Ω 17
47 Properties of probability laws All the following can be proven by decomposing a set into disjoint partitions and using the additivity and non-negativity rules. If A B, then P(A) P(B). B = A (A \ B), where A and B \ A are disjoint. So P(B) = P(A) + P(B \ A) P(A). P(A B) = P(A) + P(B) P(A B). A B=(A \ B) B. So the additivity rule gives P(A B) = P(A \ B) + P(B). Can you finish the proof? A \ B A B Ω P(A B) P(A) + P(B). P(A B C) = P(A) + P(A c B) + P(A c B c C). 17
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