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midterm2008 - Principles of Digital Communications EPFL...

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Principles of Digital Communications EPFL Summer Semester 2008 April 17, 2008 Midterm This is a closed-book exam, but a single page of hand-written notes is allowed. Cal- culators and cell-phones are not allowed. You have 2 hours to solve the problems. Try to do side calculations on a separate sheet and report well organized solutions. If we can’t read it we can’t grade it. If you use additional pages, make sure to write your name on them! Good Luck!! Name: Problem 1 / 8 Problem 2 / 12 Problem 3 / 15 Total / 35 1
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1. Problem 1 (8 points) You are playing a game of roulette. A standard roulette wheel has 37 numbers. Number 0 is colored green, and the remaining 36 numbers are equally divided between the red and the black numbers. Suppose that there are two kinds of roulette tables. When the game is played on a normal table, the ball has equal probability of landing on any of the numbers. On a loaded table, the ball lands on black numbers with 3 times more probability than on the red numbers. Otherwise, the tables are indistinguishable. The probability that the ball lands on green is the same on both tables. (a) (4 points) Let Y = Y 1 Y 2 ...Y k denote a sequence of k random outcomes of the roulette game, where Y i is the outcome of the i th game. If you are playing on a normal table, what is the probability to observe a sequence y 1 y 2 ...y k ? What is this probability if the table is loaded? (b) (4 points) Suppose you are in a city where 90% of roulettes are normal, and 10% are loaded. You go into a casino and you watch k games, observing a sequence y 1 y 2 ...y k . Find a decision rule that, with minimal probability of error, decides if this game was played on a normal or on a loaded table. Simplify your decision rule as much as possible.
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