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Random Variable Tutorial 3 STAT1301 Fall 2010 05OCT2010, MB103@HKU By Joseph Dong

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The Definition 2

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Handout Problem 1 3

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R.E. = Tossing two dice simultaneously to observe 2 digits. The possible physical outcomes are (1,1), (1,2), ……, (6,6). There are 36 of them. The State Space is the set of all physical outcomes (states): {(1,1), …, (6,6)}. 4

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The possible values of money I can win are 9, -10, and 0. The Sample Space is the set of all possible numerical outcomes I can win: {9, -10, 0} 5

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R.E. = Tossing two dice simultaneously to observe the sum of the 2 digits produced. OR R.E. = Tossing two dice simultaneously to observe the value of money you can win based on the sum of the 2 digits produced. 8

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Conclusion: Since the 3 spaces above have consistent probability measure, any one can be used as our state space or sample space, depending on your choice. They are just different representations. The consistency across the spaces are guaranteed by the defining nature of the random variable between them. Make sure you use the right probability measure for the sample/state space you work on. The choice of a good sample space is an art. A good choice of sample space—and accordingly its probability measure—can greatly simplify the solution process. 9

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Random Variable/Function 11

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Sample Space Illustrated 12

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Handout Problem 6 Hint: If you understand our discussion of Problem 1, you should immediately know an example for Problem 6. Also try to find a different kind of example. 13

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Distribution Nomenclature Cumulative Distribution Function Probability Density Function Probability Mass Function 14

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Handout Problem 3 and 4 16

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Distribution Function 17

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Handout Problem 2 Draw a graph for each question to show the random variable, the state space, the sample space, and the probability mass function on the sample space. 18

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Handout Problem 5 19

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Handout Problem 7 Hint: Try to understand “ Expectation is the coordinate of the center of mass ” 20

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