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.
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.
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.
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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