bst631-fall-2008-lecture-02

bst631-fall-2008-lecture-02 - Lecture 2 on BST 631...

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Lecture 2 on BST 631: Statistical Theory I – Kui Zhang, 08/21/2008 1 Review for the previous lecture Definition: sample space, event, operations (union, intersection, complementary), disjoint, pairwise disjoint Theorem: commutativity, associativity, distribution law, DeMorgan’s law Probability theory: sample space, sigma algebra, probability function Theorem: define a probability function on finite and countable infinite sample spaces Chapter 1 – Probability Theory Chapter 1.2 – Basic of Probability Theory 1.2.1 Axiomatic Foundations Theorem 1.2.6: Let 1 {, , } n Ss s = " be a finite set. Let B be any sigma algebra of subsets of S . Let 1 ,, n p p " be nonnegative numbers that sum to 1. For any A B , define () PAb y : i i is A PA p = . Then P is a probability function on B . This remains true if 12 {,,} s = " is a countable set. Calculating the probability of an Event: The following steps can be used to find a probability of an event from a countable sample space: 1. Define the experiment, determine the possible outcomes, and construct the sample space S . 2. Define probability function P on S .
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Lecture 2 on BST 631: Statistical Theory I – Kui Zhang, 08/21/2008 2 3. Define the event of interest A , and calculate its probability using: : () i i is A PA p = . Example 1.2.7: Example: A balance coin is tossed three times. Calculate the probability that exactly two of the three tosses results in heads. Solution: () 3 / 8 = . 1.2.2 The Calculus of Probability Theorem 1.2.8: If P is a probability function and A is any set in B , then 1. ()0 P ∅= ; 2. () 1 ; 3. 1 ( ) c =− . Example: Suppose a pair of fair dice is rolled. What is the probability that the sum of face values is at least 4? Solution: ( ) 33/36 11/12 == . {( , ):1 6,1 6} Si j i j =≤ . {( , ) : ( , ) , and 3} {(1,1), (1, 2), (2,1)} c BA i j i j S ij + = .
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Lecture 2 on BST 631: Statistical Theory I – Kui Zhang, 08/21/2008 3 Theorem 1.2.9: If P is a probability function and A is any set in B , then 1. () ( ) ( ) c PB A PB PA B ∩= ; 2. ( ) () ( ) PA PB ∪= + ; 3. If A B , then PA . Note: The use of Venn Diagram can help you visualizing a situation but not constitute a formal proof. Example 1.2.10 (Bonferroni’s Inequality): Let A and B be two events such that P ( A ) = P ( B ) = 0 . 95 . What can we say about the probability that both events will occur? Solution: ( ) ) ()1 0 . 9 5 0 . 9 51 0 . 9 0 =+− ≥+− = + = . Theorem 1.2.11: If P is a probability function, then 1. 1 ( ) i i PA C = =∩ for any partition 12 ,, CC " of the sample space S (law of total probability); 2. 1 1 ( ) ii i i P A = = for any sets AA " (Boole’s Inequality).
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This note was uploaded on 03/17/2010 for the course STATISTIC 472 taught by Professor Amjad during the Spring '08 term at Yarmouk University.

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bst631-fall-2008-lecture-02 - Lecture 2 on BST 631...

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