Measure Theory

Measure Theory - Economics 245A Introduction to Measure...

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Economics 245A Introduction to Measure Theory The goal of this lecture is to take the axioms of probability, which are intro- duced as the basis for statistical theory, and relate them to measure theory. Probability Probability is a subject that can be studied independently of statistics, it forms the foundation for statistics. For example, what is the probability that a head comes up twice in a row if we toss an unbiased coin? The answer, .25, is calculated without need of statistical inference. Kolmogorov (1933) related probability to the concept of measure in integration theory. In so doing he exploited analogies between set theory and the concept of a random variable and developed the axiomatic theory of probability. Axiomatic Theory of Probability remain vague until they are illustrated. Random experiment . An experiment that can be repeated under identical con- ditions for which all possible outcomes of the experiment are known beforehand, although on any trial the realized outcome is not known beforehand. Sample space. The set of all possible outcomes of a random experiment. Simple event. An event that cannot be a union of other events. Composite event. An event that is not a simple event. Example 1. Random experiment. Tossing a coin twice. Sample space: f ( H; H ) ; ( H; T ) ; ( T;H ) ; ( T;T ) g . The (composite) event that at least one head occurs: ( H; H ) [ ( H; T ) [ ( T; H ). Example 2. Random experiment. Reading the temperature (F) at UCSB at noon on No- vember 1. Sample space: real interval (0,100). Events of interest are intervals contained in the sample space. A probability is a nonnegative number we assign to every event. The axioms of probability are the rules we agree to follow when we assign probabilities. Axioms of Probability 1
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Left Header Right Header (1) P ( A ) 0 for any event A . (2) P ( S ) = 1, where S is the sample space. (3) If f A i g , i = 1 ; 2 ; :: : , are mutually exclusive (that is, A i \ A j = ; for all i 6 = j ), then P ( A 1 [ A 2 [ : :: ) = P ( A 1 ) + P ( A 2 ) + : :: . The third rule is consistent with the frequency interpretation of probability, for relative frequency follows the same rule. If, at the roll of a die, A is the event that the die shows 1 pip and B is the event that the die shows 2 pips, the relative frequency of A [ B is the sum of the relative frequencies of A and B . We want probability to follow the same rule. If the sample space is discrete, as in example 1, it is possible to assign proba- bility to every event (that is, every possible subset of the sample space) in a way that is consistent with the probability axioms. If the sample space is continuous, however, as in example 2, it is not possible to do so. In such a case we restrict attention to a smaller class of events to which we can assign probabilities in a manner consistent with the axioms.
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This note was uploaded on 12/26/2011 for the course ECON 245a taught by Professor Staff during the Fall '08 term at UCSB.

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Measure Theory - Economics 245A Introduction to Measure...

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