MATH 632 Final: 2004 Math 632 - Spring Final Exam
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Instructions: show calculations and give concise justi cations for full credit. Points add up to 200: four cages labeled a, b, c and d are connected by tubes as indicated in the diagram below. A rat runs from one cage to the other. Let xn denote the nth cage the rat visits. Suppose the rat starts in a, so x0 = a. (a) (30 pts) what is the expected number of cages the rat visits until its. Rst return to cage a, not counting the rst and last visits to a ? (b) (25 pts) find lim n . P {xn = a, xn+1 = b, xn+2 = a} in other words the long-term limiting probability that we see the rat visit. A b a: (a) (25 pts) consider the following game. Pick a number k between 1 and 6. Roll a fair six-sided die repeatedly until you get the rst k, and add up the numbers you got.
