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hw07solr - University of Illinois Spring 2010 ECE 313...

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University of Illinois Spring 2010 ECE 313: Problem Set 7: Solutions Decision Making; Independent Events; System Reliability 1. [ Mutually exclusive events ] Let C = A ! B ( ) c denote the event that neither A nor B occurs on a trial of the experiment, and notice that one of the three events A, B, and C always occurs on a trial. Then, A occurs before B does if for n = 1; 2; 3, … , C occurs on the 1st, 2nd, … , (n - 1)- th trials and A occurs on the n-th trial. By independence of the trials, P A before B { } = P A ( ) + P C ( ) P A ( ) + P C ( ) ! " # $ 2 P A ( ) + ... = P A ( ) 1 % P C ( ) = P A ( ) P A ( ) + P B ( ) Thus, P Bbefore A { } = 1 ! P A before B { } = P B ( ) P A ( ) + P B ( ) . The way to think about this is we can ignore all trials on which C occurs. On the very first trial on which one of A and B occurs, what are the chances that A occurs? Thus, P A| A ! B ( ) = P A ( ) P A ! B ( ) = P A ( ) P A ( ) + P B ( ) . 2. [ Detection problem with geometric distribution vs. Poisson distribution ] (a) Mean Variance Standard deviation H 0 λ = 10 λ = 10 3.162 H 1 1/p = 10 (1- p)/p 2 = 90 9.487 Note: The means are the same but the distribution under H 1 is more spread out (e.g. its standard deviation is three times larger) than the distribution under H 0 . Moreover, on one hand, the geometric pmf with p = 10 is slowly decreasing, with values p(1) = 0.1, p(10) = 0.0387, p(20) = 0.0135; and p(30) = 0.00471, for example. On the other hand, the Poisson pmf for λ = 10 has a peak value at k = 10 and falls off rapidly away from k = 10. (b) The ML rule is to decide H1 is true if
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hw07solr - University of Illinois Spring 2010 ECE 313...

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