chap03PRN

chap03PRN - Chapter 3.1 Random Experiment, Outcomes and...

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Unformatted text preview: Chapter 3.1 Random Experiment, Outcomes and Events In the analysis of economic data any conclusions involve some uncertainty. To incorporate ideas of uncertainty into the study of economic data an understanding of probability theory is needed. A random experiment has a set of basic outcomes . Example: View the random experiment as investing in the stock market. For a selected company the possible outcomes for the daily change in the closing price of one share are: increase 1 O decrease 2 O no change 3 O Clearly, at the start of the business day, there is uncertainty as to which outcome will occur. The sample spaceS is the set of all basic outcomes. An event is a subset of basic outcomes from the sample space S . Example: For the daily change in stock market price define the events: no loss ]O,O[ A 3 1 no gain ]O,O[ B 3 2 1E con 325 Chapter 3 Consider two events A and B . The intersection is denoted by B A . This is the set of basic outcomes in the sample space S that belong to both A and B . If events A and B have no common basic outcomes then is the empty set. In this case A and B are mutually exclusive . B A The union is denoted by . This is the set of basic outcomes that are in either A or B or both. B A If the union of several events gives the whole sample space S then the events are collectively exhaustive . The complement of is denoted by A A . This is the set of basic outcomes in the sample space S that do not belong to the event . Therefore: A S A A and A A are collectively exhaustive, and A A empty, and A A are mutually exclusive 2E con 325 Chapter 3 A Venn diagram gives an illustration. Sample Space S B A B A In the next Venn diagram the events A and B are mutually exclusive. Sample Space S 3E con 325 Chapter 3 A Venn diagram for the complement of event A . Sample Space S A As an alternative to the above, the next Venn diagram demonstrates that and A A are collectively exhaustive and mutually exclusive. Sample Space S A A 4E con 325 Chapter 3 Some Results Consider two events A and B . The events and B A B A are mutually exclusive, and B)B A()B A( . That is, the union is B . The events and A B A are mutually exclusive, and B A)B A( A . That is, the union is . B A These results are shown with a Venn diagram. Sample Space S B A B A 5E con 325 Chapter 3 A General Result Let,, . . . , be K mutually exclusive and collectively exhaustive events. That is, 1 E 2 E K E is the empty set for all j i E E ji , and S E E E K 2 1 6 Suppose is some other event in the sample space S . Then the K events: A A E 1 ,, . . . , A E 2 A E K are mutually exclusive and their union is . A This result is shown in a Venn diagram with K = 4. 1 E 2 E 3 E 4 E Sample Space S A E 1 A E 2 A E 3 A E 4 Event A 6E con 325 Chapter 3 Chapter 3.2 Probability For a random experiment probabilities can be assigned to outcomes and events. Denote: )O(P i the probability that basic outcome...
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chap03PRN - Chapter 3.1 Random Experiment, Outcomes and...

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