01-intro-sets-axioms

01-intro-sets-axioms - IE 111 Fall Semester 2007 Note Set#1...

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IE 111 Fall Semester 2007 Note Set #1 Introduction This course will cover the fundamentals of probability theory. It is essentially a mathematics course (indeed the Accreditation Board of Engineering and Technology, which accredits engineering schools, classifies this course as a math course, not an engineering course. It is the only course in Industrial Engineering classified as "math"). Probability is extremely useful in science and engineering. That is to say, it has many useful applications. It is my hope that you will see why it is useful, and how it can be used. It is also true that in this class, many of the examples are quite simple. For example, a probabilistic model of flipping a fair coin (fair means 50% chance for heads, 50% for tails) is used over and over. While quite simple, this model serves as a building block for increasingly more complex (and useful) models. Many other courses in Industrial Engineering rely on basic knowledge of probability and thus build on the fundamentals learned in this class. Why is probability important? Many real world processes and phenomena exhibit random or "uncertain" behavior. None-the-less, these processes and phenomena do exhibit a certain amount of predictability . The ability to use probability theory to predict the future outcomes of random processes and phenomena is indeed quite useful. The other use of probability is to quantify the amount of randomness/uncertainty exhibited by the process or phenomena. Probability theory is also a prerequisite for learning statistics. Toward the end of this course we will also learn some introductory statistics. Statistics and probability are quite different subjects . Statistics is the science of analyzing and drawing inference from DATA , the key word being data. Statistics is an inferential process whereby data is drawn from a population, then "statistics" is applied to the data to make inferences and draw conclusions about the population as a whole. Thus statistics is inherently INDUCTIVE . In probability theory, there is no data to analyze. We start with an assumed model of how some process or phenomenon behaves, and then apply probability theory to DEDUCE other properties and behaviors. The point is that probability is deductive. Examples of the use of statistics 100 people are randomly selected from the American population. They are asked whom they will vote for in the election, candidate A or B. 48 indicate "A" and 52 indicate "B". Estimate the probability that "B" will win. More importantly, determine the margin of error of your estimate . Given the monthly unemployment percentages for the last 48 months, estimate next month’s unemployment percentages. Also find a range within which next month’s unemployment percentages will lie with 95% probability (e.g. there is a 95% chance that next month’s unemployment will lie in the range 4.0% to 6.2%) You are given survival time data on two groups of heart patients. One group regularly took a
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new cholesterol-reducing drug, while the other took placebos. Test whether or not there is a significant difference in the mean survival time of the two groups.
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