Lecture+2+Probability

Lecture+2+Probability - ECON 123A, Fall 2011, Lecture 2 2-1...

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ECON 123A, Fall 2011, Lecture 2 Dale J. Poirier 2-1 Lecture 2 REVIEW OF PROBABILITY “Probability is the very guide to life.” Cicero, De Natura Deorum (45BC) In contrast: “Probability does not exist.” de Finetti (1974)
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ECON 123A, Fall 2011, Lecture 2 Dale J. Poirier 2-2 Definition: An experiment is a process whose outcome is not known in advance with certainty. The sample space , denoted by s , is the set of all possible outcomes of the experiment under consideration. An event is a subset of the sample space. C “Probability” is defined over appropriate subsets of the sample space. C Frequentists define the probability of a particular outcome of an experiment to be the proportion of the time that the outcome occurs in the long run. More formally we have the following definition.
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ECON 123A, Fall 2011, Lecture 2 Dale J. Poirier 2-3 Definition (Objective or Frequentist): Let N be the number of trials of an experiment, and let m(N) be the number of occurrences of an event A in the N trials. Then the probability of A is defined to be (assuming the limit exists): C Adherents to this definition think of “probability” as a property of reality.
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ECON 123A, Fall 2011, Lecture 2 Dale J. Poirier 2-4 C Frequentists argue that situations that do not admit repetition under essentially identical conditions are not within the realm of statistical enquiry. C Therefore, the frequentist /objectivist interpretation cannot be applied to: B unique, once-and-for-all type of phenomenon (e.g., elections), B theories (e.g., “monetarist” or “Keynesian” economics), or B uncertain past events (e.g., whether the Cleveland Indians won the 1954 World Series). C Other interpretations of probability (to be studied) permit such everyday applications.
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ECON 123A, Fall 2011, Lecture 2 Dale J. Poirier 2-5 B.1 Random Variables and Their Probability Distributions “A random variable is the soul of an observation. . .. An observation is the birth of a random variable.” Watts (1991, p. 291) C The sample space is tedious to work with if its elements cannot be directly manipulated mathematically. Definition: A random variable is a “well-behaved” mapping (i.e., function) from the sample space s onto the set U of real numbers. http://www.math.uah.edu/stat/applets/DiceExperiment.xhtml
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ECON 123A, Fall 2011, Lecture 2 Dale J. Poirier 2-6 C A random variable is a numerical summary of an experimental outcome. C Random variables can be either discrete, continuous, or both. B Discrete random variables take on only a finite or countable number of values. < A discrete random variable that takes on only two values (usually 0 and 1) is called a Bernoulli random variable. B Continuous random variables take on a continuum (uncountable number) of possible values.
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ECON 123A, Fall 2011, Lecture 2 Dale J. Poirier 2-7 Definition: The probability mass function (pmf) of a discrete random variable is the list of all possible values of the variable and the probability that each value will occur.
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This note was uploaded on 12/13/2011 for the course ECON 123a taught by Professor Staff during the Fall '08 term at UC Irvine.

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Lecture+2+Probability - ECON 123A, Fall 2011, Lecture 2 2-1...

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