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20082ee131A_1_HW4SOL

20082ee131A_1_HW4SOL - EE 131A Homework#4 Solution Spring...

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EE 131A Homework #4 Solution Spring 2008 K. Yao 1. We know A B = φ. (a) If A and B are independent, Pr ( A ) Pr ( B ) = Pr ( A B ) = 0 because A B = φ. Therefore Pr ( A ) = 0 or Pr ( B ) = 0 or both equal 0 . . (b) If Pr ( A ) = 0 or Pr ( B ) = 0 or both equal to 0, we have Pr ( A ) Pr ( B ) = 0 . But Pr ( A B ) = 0 because A B = φ. Therefore Pr ( A B ) = Pr ( A ) Pr ( B ) and A and B are independent. 2. Let hypothesis H k denote that a coin of type k was drawn. Its prior probability is 1 9 . Bayes rule tells us that the posterior probability of hypothesis H k is Pr ( H k | 7 H out of 25) = C 25 7 π 7 k (1 - π k ) 18 9 m =1 C 25 7 π 7 m (1 - π m ) 18 . Since the denominator in the above expression does not dependent on k, we only need to consider the maximiation of p m (1 - p ) n - m with respect to p where p = m/n. In our case, we know maximization occurs at m/n = 7 / 25 = 0 . 28 . This shows we should decide on H 3 or coin type 3 since π 3 = 0 . 3 . 3. X is a random variable with a binomial distribution with parameter n and p = 0 . 9 a. Prob. the system will declare an airplane = P (1 X 2) = 1 - P ( X = 0) = 1 - 2 0 (0 . 9) 0
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