01 Random Variables

01 Random Variables - Economics 140A Random Variables...

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Economics 140A Random Variables Probability In everyday usage, probability expresses a degree of belief about an event or statement with a number between 0 and 1. This de&nition of probability is sub- jective in nature because di±erent individuals assign di±erent probabilities to an event. We work with an objective de&nition of probability. Our objective de&ni- tion of the probability that an event occurs is given by the limit of the empirical frequency of the event as the number of replications of the experiment, from which di±er across individuals. Probability is a subject that can be studied independently of statistics; it forms the foundation of 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. Axioms of Probability remain vague until they are illustrated. Sample space . The set of all the possible outcomes of an experiment. Event . A subset of the sample space. Simple Event. An event that cannot be a union of other events. Composite Event . An event that is not a simple event. Example 1. Experiment. Tossing a coin twice. Sample space: f HH, HT, TH, TT g . The event that a tail occurs at least once: HT [ TH [ TT. Example 2. Experiment. Reading the temperature (F) at UCSB at noon on November 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. (Often, Venn diagrams are used to determine relations among the probabilities assigned to di±erent sets.) Axioms of Probability
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(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 prob- ability 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 con- tinuous, 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. For example, the class of all intervals
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This note was uploaded on 09/04/2011 for the course ECON 140a taught by Professor Staff during the Fall '08 term at UCSB.

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01 Random Variables - Economics 140A Random Variables...

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