2.8 Interpretations of probability: One abstract way to define probability is via measure theory, in that Pr(-) is simpl

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2 8 Interpretations Of Probability One Abstract Way To Define Probability Is Via Measure Theory In That Pr Is Simpl 1 (67.43 KiB) Viewed 36 times

2.8 Interpretations of probability: One abstract way to define probability is via measure theory, in that Pr(-) is simply a "measure" that assigns mass to various events. For example, we can "measure" the number of times a particular event occurs in a potentially infinite sequence, or we can "mea- sure" our information about the outcome of an unknown event. The above two types of measures are combined in de Finetti's theorem, which tells us that an exchangeable model for an infinite binary sequence Y₁. Y₂,... is equivalent to modeling the sequence as conditionally i.i.d. given a pa- rameter 0, where Pr(0 < c) represents our information that the long-run frequency of 1's is less than c. With this in mind, discuss the different ways in which probability could be interpreted in each of the following scenarios. Avoid using the word "probable" or "likely" when describing probability. Also discuss the different ways in which the events can be thought of as random. a) The distribution of religions in Sri Lanka is 70% Buddhist, 15% Hindu, 8% Christian, and 7% Muslim. Suppose each person can be identified Exercises 227 by a number from 1 to K on a census roll. A number z is to be sampled from {1..... K} using a pseudo-random number generator on a computer. Interpret the meaning of the following probabilities: i. Pr(person z is Hindu); ii. Pr(z = 6452859); iii. Pr(Person z is Hindulz-6452859). b) A quarter which you got as change is to be flipped many times. Inter- pret the meaning of the following probabilities: i. Pr(0, the long-run relative frequency of heads, equals 1/3): ii. Pr(the first coin flip will result in a heads); iii. Pr(the first coin flip will result in a heads | 0 = 1/3). c) The quarter above has been flipped, but you have not seen the out- come. Interpret Pr(the flip has resulted in a heads).