stat_module_4 - Advanced Probability and Statistics Module...

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Advanced Probability and Statistics Module 4 Aloha, stat people. This problem set focuses primarily on probability and its application to statistics. This is the really cool stuff. It corresponds to chapters 7 and 8 in the book. There are a few topics from these chapters that I’ll cover in the next module. I’ll supplement the book with some additional material and examples. I’ll also try to lay a more rigorous foundation for probability than the book does by incorporating some set theory, much of which you already know. Here we go. Here’s an example of a random experiment: roll a die (singular of dice) and see how many spots are on top of it when the die stops rolling. I agree, it’s a boring example, but it’s simple and familiar. Anyway, it’s a random experiment since there is no way to determine the outcome before the roll. One outcome is, of course, 5 spots. The sample space, S , of the experiment is the set of all possible outcomes. Here, S = {1, 2, 3, 4, 5, 6}. An event is any subset of S (any collection of outcomes). The event E 1 = {1, 3, 5} is the event that an odd number of spots faces up on the die. Note that E 1 S , as required by the definition. The event E 2 = {6} is called a simple event since it consists of just one outcome. 1. How many simple events are there for this situation? 2. There are  2 6  = 15 different events with exactly two outcomes since this is the number of ways two outcomes can be  chosen from six. List a few of them. Pick one and state exactly what it means, even though it may seem obvious. 3. Calculate, separately, the number different events with 3, 4, 5, and 6 outcomes. 4. If asked to list the all the events that contain no outcomes at all, you’d have to find a subset of  S  with zero elements.  This would be the empty set, { }. (The empty set is a subset of every set.) Counting the empty set, how many total different  events are there?  Hint: the answer should equal 2 6 5. It is no coincidence that the last answer was two to the power of the number of elements in  S . In general, the number  of subsets of a set with  n  elements is 2 n . Curious, huh? An explanation is forthcoming. For now, how many subsets are  there of the set containing the letters of the alphabet? 6. Demonstrate that this trick works with the set  L  = { a b c } by listing and counting all subsets. Do is systematically by  listing and counting all the zero-element, one-element, two-element, and three-element subsets. 7. When counting the subsets in the last question, you should notice that those four numbers appear in the third row of a  special triangle (where the pinnacle is row zero). Dag gone, is that cool, or what?! This is no coincidence either. Use the 
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