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Unformatted text preview: Lecture 1. Probability Spaces and Models Before beginning this lecture, write answers to the following questions. Obviously, you will struggle with some of them, but your attempts to provide answers will help you focus on the important ideas in this lecture. 1. What is a probability space? A discrete probability space? 2. What is a probability mass function (pmf)? What is the difference between a pmf and a probability measure? How is a pmf used to define a probability measure? 3. What is the difference between a probability space and a probability model? How are the words experiment, outcome, event used in probability modeling? 4. Give an example of a probability model. Introduction. Let X be a set. A probability measure on X is an assignment of numerical values to some of the subsets of X . The sets to which values are assigned are called measurable . If E X is measurable, its assigned probability measure is denoted P ( E ). We call P ( E ) the probability of E . A set equipped with a probability measure is called a probability space . Probability theory is about probability spaces and how to use them to model situations that involve chance or uncertainty. The assignment P ( ) must satisfy certain technical conditions. We will describe these in detail later on, as needed. To give you a sense of what the requirements are, let me just say that the following are demanded: P ( ) = 0 and P ( X ) = 1; P ( E ) 1; If E 1 , E 2 , . . . are disjoint, then P parenleftBig uniontext i =1 E i parenrightBig = i =1 P ( E i ). In this course, we often use the symbol (the upper case Greek letter omega) to denote a probability space. Lower case omegawhich is written will be used as a variable to denote an unspecified element of . Other lower case Greek letters such as (alpha), (beta), (gamma) and (delta) may be used to denote specific elements of . 1.1. Discrete probability spaces. For the first several weeks of this course, we will be working mainly with one special kind of probability space. Definition. 1 A discrete probability space is a finite or countable 2 set equipped with a probability mass function (or pmf, for short). A pmf on is a function f : [0 , 1] such that f ( ) = 1. The number f ( ) is called the mass assigned to . Given a pmf, we define a probability measure P by the rule: if A , then P ( A ) := summationdisplay a A f ( a ) . (1) 1 In mathematics, a definition is a statement that introduces a new terminology or nota- tion. Definitions need to be read with great care, remembered and referred to whenever the terminology or notation in them is used....
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This note was uploaded on 11/29/2011 for the course MATH 3355 taught by Professor Britt during the Spring '08 term at LSU.
- Spring '08