bst631-fall-2008-lecture-01

bst631-fall-2008-lecture-01 - Lecture 1 for BST 631...

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Lecture 1 for BST 631: Statistical Theory I – Kui Zhang, August 19, 2008 1 Chapter 1 – Probability Theory Chapter 1.1 – Set Theory Definition: A set is a collection of finite or infinite elements where ordering and multiplicity are generally ignored. Definition 1.1.1: The set, S , of all possible outcomes of a particular experiment is called the sample space for the experiment. Examples: Consider the following experiments: Example: toss a coin once. What is the sample space? Solution: {,} SH T = . Example: toss a coin 2 times. What is the sample space? Solution: {( , ),( , ),( , ),( , )} H H T T H T T = (ordered). Solution: { (,) , ( ,) , ( } S H HT T = (unordered). Example: roll a dice twice. What is the sample space? Solution:
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Lecture 1 for BST 631: Statistical Theory I – Kui Zhang, August 19, 2008 2 Example: roll the dice until 1 appears and record the number of times that the dice has been rolled. What is the sample space? Solution: {1, 2, 3, } S = " . Example: record the reaction time to a certain stimulus in an experiment. What is the sampling space? Solution: (0, ) S =∞ . Example: Choose a point in the interval [0, 1]. What is the sample space? How about choose a point from the square bounded by the points (0, 0), (0, 1), (1, 0), and (1, 1)? Solution: [0,1] S = and {( , ):0 1,0 1} Sx y x y =< < < < . Definition: A sample space is countable if its elements can be put into 1-1 corresponding with a subset of integers. Types of the Sample Space: Countable: finite, infinite Uncountable: infinite Definition 1.1.2: An event is any collection of possible outcomes of an experiment, i.e., any subset of S including S itself. Note: An event A is said to occur if the outcome of the experiment is in the set A .
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Lecture 1 for BST 631: Statistical Theory I – Kui Zhang, August 19, 2008 3 Two Relationships defining order and equality: 1. A Bx A x B ⊂⇔ ∈⇒ (containment, i.e., A is a subset of B ) 2. and A BA B B A =⇔⊂ (equality) Elementary Set Operations: 1. Union: {: o r } A x Ax B ∪= , i.e., set of elements that belong to A or B or both 2. Intersection: { : and } A x A x B ∩= i.e., set of elements that belong to both
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bst631-fall-2008-lecture-01 - Lecture 1 for BST 631...

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