# Chapter 2 - Chapter 2 Foundation of Probability Theory...

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Chapter 2 Foundation of Probability Theory Abstract: Probability theory is the foundation of statistical science, providing a mathematical means of modeling random experiments or uncertainty. Through these mathematical models, researchers are able to draw inferences about the random experiments using observed data. The aim of this chapter is to outline the basic ideas of probability theory that are fundamental to the study of statistics. The entire structure of probability, and therefore of statistics, can be built on the relatively straightforward foundation given in this chapter. Key words: Random experiment, Sample space, Event, Borel ﬁeld, Sigma algebra, Probability function, Probability space, Sets, Union, Intersection, Complement, Permutation, Combination, Bayes’ theorem, Conditional probability, Independence, Multiplication rule. 2.1 Random Experiments Many kinds of scientiﬁc research may be characterized in part by the fact that repeated experiments, under more or less the same conditions, are standard practice. For instance, an economist may be concerned with the prices of three speciﬁed commodities at various time periods. The only way in which a researcher can elicit information about such a phenomenon is to perform repeated experiments. Each experiment terminates with an outcome. But it is characteristic of these experiments that the outcome cannot be predicted with certainty prior to the performance of the experiment, although the experiment is of such a nature that a collection of every possible outcome can be described prior to its performance. Deﬁnition 1 (2.1) . [Random Experiment]: A random experiment is a mechanism which has at least two possible outcomes, and which outcome to occur is unknown in advance. In other words, a random experiment is a mechanism for which the outcome cannot be predicted with certainty. Remarks: The word “experiment” here means a process of observation or measurement in a broad sense. It is not necessarily a real experiment as encountered in (e.g.) Physics. There are two essential elements of a random experiment: the set of all possible outcomes; the likelihood with which each outcome will occur. As is well-known, modern economics is a study on resource allocation in an uncertain envi- ronment. When an economic agent makes a decision, he or she usually does not know precisely the outcome of his or her action, which usually will arise in an uncertain manner. This, to some extent, is similar to a random experiment. Modern econometrics is built upon the following two fundamental axioms: 1

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An economic system can be viewed as a random experiment governed by some probability law. Any economic phenomenon (often in form of data) can be viewed as an outcome of this random experiment. The random experiment is usually called a “data generating process”.
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## This note was uploaded on 01/07/2012 for the course ECON 6190 taught by Professor Hong during the Fall '07 term at Cornell University (Engineering School).

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Chapter 2 - Chapter 2 Foundation of Probability Theory...

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