HW3 - 1 was sent. b) Is this rule the MAP rule for any...

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1 EE 7615 Problem Set # 3 Due: 10,3,11 1) Problem 4.3 of Chapter 4, page 266. 2) Find the optimum decision regions for a binary hypothesis testing problem with observation R , where the hypotheses are equally likely and the conditional distribution of R given each hypothesis is given by Given H 0 : R is uniformly distributed over [ - 2 , 2] . Given H 1 : R is Gaussian with mean 0 and variance 1 . Find the probability of error. 3) The receiver in a binary communication system receives R where R = N when M = 0 is sent R = N + 4 when M = 1 is sent, where N is a Laplacian random variable with p N ( x ) = 1 2 e -| x | , - ∞ < x < and N is independent of the message M . Consider the decision rule g ( r ) = b 0 r < 1 1 r 1 a) For this rule ±nd P ( E | 1) , i.e., the probability of error given that message
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Unformatted text preview: 1 was sent. b) Is this rule the MAP rule for any prior probabilities on the message M ? If so specify the probability p = P ( M = 0) for which this is the MAP rule. If not, explain why not. 4) Consider a binary hypothesis testing problem where the observation is R and its conditional distribution is given by Given H : p R | H ( r ) N (1 , 4) Given H 1 : p R | H 1 ( r ) N (-1 , 1) a) Show that the optimal decision rule is equivalent to comparing a function of the form ar 2 + br with a threshold. b) Find a and b when p = P ( H ) = 1 / 3 ....
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