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# STT - STT 861 Theory of Probability and Statistics I 0...

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STT 861: Theory of Probability and Statistics I 0 Introduction According to Wikipedia, statistics is a mathematical science pertaining to the collection, analysis, interpretation or explanation, and presentation of data. It has applications to a wide variety of academic disciplines, from the natural and social sciences to the humanities, government and business. Since the quantities that we observe are most of the time random we need a theory about the likelihood of potential events and the underlying mechanics of random systems. This is what probability theory is all about. In this sequence of two courses (STT 861 and 862) we will first study the basic theory of probability and random variables and then use that to learn about statistical methods. 1 Probability 1.1 Sample Spaces and Events Probability is a model that assigns to each event A a number, P ( A ), de- scribing the likelihood of this event to occur. An event is just a collection of possible outcomes, and the collection of all possible outcomes (the biggest event) is called the sample space , often denoted by Ω. Example 1.1 (Tossing a coin) . The sample space is Ω = { H, T } and the event that “the outcome is head” is { H } . Example 1.2 (Throwing a die) . The sample space is Ω = { 1 , 2 , 3 , 4 , 5 , 6 } and the event that “the outcome is even” is { 2 , 4 , 6 } . Example 1.3 (Driving to work) . When Lisa drives to work, she passes through three intersections with traffic lights. At each light, she either stops, s , or continues, c . In this example, the sample space, Ω = { ccc, ccs, css, csc, sss, ssc, scc, scs } , “she does not stop at all”= { ccc } , “she has to stop in all the lights’= { sss } , “she stops in at most one light”= { ccc, ccs, csc, scc } and so on. More examples are available in the textbook. Please read them. 1

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1.2 Probability Measure Probability measure (or simply probability) is a map that assigns a real number P ( A ) to each event A . It must form a consistent model, as described in the following set of requirements (axioms): 1. For every event A , P ( A ) 0. 2. For the sample space (biggest event) Ω, P (Ω) = 1. 3. For pairwise disjoint (mutually exclusive) events A 1 , A 2 , . . . , P [ i =1 A i ! = X i =1 P ( A i ) . As a visual aid, mentally interpret probabilities as areas with the use of Venn diagrams. Please read the textbook for a quick review of set theory and Venn diagrams. More specifically, read the commutative laws, the asso- ciative laws, and the distributive laws of the set operations. There are two more laws called de Morgan’s Laws : 1. ( A B ) c = A c B c , and 2. ( A B ) c = A c B c . Example 1.4 (An easy example) . 16 universities are taking part in a bas- ketball tournament. In the first round there are 8 games (1 vs. 2, 3 vs. 4, . . . , 15 vs. 16). An outcome is a list of winners, e.g. (2 , 3 , 5 , 7 , 9 , 12 , 13 , 16). An event is a set of several outcomes; the sample space has 2 8 = 256 out- comes. Setting how likely each outcome is, we can compute P ( A ) by adding up the likelihoods of the outcomes comprising the event A .
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