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Unformatted text preview: THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT1306 INTRODUCTORY STATISTICS (SEMESTER 1 2008/2009) 2 Probability To most people, probability is a loosely de&ned term employed in everyday¡s conversation to indicate the measure of one¡s belief in the occurrence of a future event. Say, what is the chance of arriving on time if one takes the bus? takes the taxi? However, for scienti&c purposes, it is necessary to give the word probability a de&nitive, clear interpretation. One can think of probability as the language in discussing uncertainty. For instance, in tossing a coin (random experiment), the outcomes can either be f H g or f T g . If we know exactly what the outcome of the next trial will be, then we are certain that, say, a f H g will show up next. However, it is impossible to know exactly what is going to happen in reality - uncertainty. De&nition. A random experiment is a process leading to at least two possible outcomes with uncertainty as to which will occur. Examples are (i) tossing a coin; (ii) rolling a dice; (iii) daily changes in Hang Seng Index. De&nition. The possible outcomes of a random experiment are called the basic outcomes , and the set of all basic outcomes is called the sample space , S . Examples are (i) S = f H;T g ; (ii) S = f 1 ; 2 ; 3 ; 4 ; 5 ; 6 g ; and (iii) S = f up, down g . De&nition. An event is a set of basic outcomes, or a collection of some basic outcomes from the sample space, and it is said to occur if the random experiment gives rise to one of its constituent basic outcomes. Example 2.1. In a random experiment of throwing two dices, the interest falls in the total of the numbers turned up by the two dices. Therefore, the basic outcomes are 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 and the sample space is S = f 2 ; 3 ; 4 ; 5 ; 6 ; 7 ; 8 ; 9 ; 10 ; 11 ; 12 g . Let A , B and C be 3 events where A = f even g = f 2 ; 4 ; 6 ; 8 ; 10 ; 12 g ; B = f odd g = f 3 ; 5 ; 7 ; 9 ; 11 g ; and C = f sum = 8 g = f (2 ; 6) ; (3 ; 5) ; (4 ; 4) ; (5 ; 3) ; (6 ; 2) g . De&nition. The probability P ( E ) of an event E 2 S is a set function de&ned on a class of all events satisfying the axioms of probability which will be discussed later. Theorem 2.1. If an experiment is repeated for n times, and if the outcomes do not a/ect each other. Let r be the number of occurrences of event A . Then, P ( A ) = r=n as n ! 1 where P ( A ) is the long-run frequency. 1 From the frequentist&s view of probability, the frequency interpretation of prob- ability is a long-run average. More precisely, if the random experiment can be repeated over and over again with n = total number of trials and r = number of trials in which event A actually happened out of the n trials. Then, P ( A ) is simply the proportion or rate of occurrences of event A in the long run; or P ( A ) = r n = relative frequency of event A ....
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