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132A-WQ03 - Math 132a Stochastic Processes Final Exam Write...

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Math 132a: Stochastic Processes Final Exam (3/22/03) Write your name and student ID number in the upper right-hand corner of this sheet and write your initials on each page of your exam. Each problem is worth the same number of points. You must justify or defend your answer for each problem. 1. Suppose that you flip a fair coin four times. Let A 1 be the event that the first coin flips 1 and 2 agree, let A 2 be the event that coin flips 2 and 3 agree, let A 3 be the event that coin flips 3 and 4 agree, and let A 4 be the event that coin flips 4 and 1 agree. Are these four events pairwise independent? Are they mutually independent? 2. In the computer game Minesweeper, you have a field of squares, some of which contain mines. When you step on a square which is not a mine, it tells you the number of neighboring mines. Suppose that at the beginning each square in the game has a 1 4 chance of being a mine and that all of these probabilities are independent. Suppose that later in the game you have three squares A , B
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