sol12 - U.C. Berkeley CS70: Discrete Mathematics and...

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Unformatted text preview: U.C. Berkeley CS70: Discrete Mathematics and Probability Problem Set 12 Lecturers: Umesh Vazirani & Christos H. Papadimitriou Due November 29, 2006 at 4:00pm Problem Set 12 Solutions 1. Total Annihilation [10 points] Each missile hits each silo with equal probability 1 2620 . Hence T i , the number of trials to destroy the i th missile once ( i- 1)th have been destroyed, has geometric distribution with success probability n i +1 n for n = 2620. Hence, the expected total number of trials is: E [ T ] = E [ n summationdisplay i =1 ] = n summationdisplay i =1 n n- i + 1 = = n summationdisplay j =1 n j n log n = 2620log 2620 = 20622 Notice this is exactly the coupon collector problem discussed in class. 2. Significance Levels [10 points] a) m = 70 + 8 + 10 + 13 + 14 + 16 + 17 + 26 8 = 83 v = 13 2 + 5 2 + 3 2 + 0 + 1 2 + 3 2 + 4 2 + 13 2 7 56 . 86 Hence, the estimate for the standard deviation is v 7 . 54....
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sol12 - U.C. Berkeley CS70: Discrete Mathematics and...

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