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 S s 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 anyA B , define ( ) P A by : ( ) i i i s A P A p = . Then P is a probability function on B . This remains true if 1 2 { , , } S s 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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