Mathematics for Computer Science
17.8. Mutual Independence 733 The guard will truthfully tell Voldemort the name of one of the prisoners to be released. We’re in ...
Chapter 17 Conditional Probability734 LetDbe the event thatan even number ofcoins are heads. (a)Use the four step method to de ...
17.8. Mutual Independence 735 Problems for Section 17.8 Exam Problems Problem 17.26. Sally Smart just graduated from high school ...
Chapter 17 Conditional Probability736 (d)Show that the event that Sally attends Yaleis notindependent of the event that she is h ...
17.8. Mutual Independence 737 satisfy the “product rule.” That is, PrŒA\B\CçDPrŒAçPrŒBçPrŒCç; but arenotmutually independe ...
Chapter 17 Conditional Probability738 (d)What is the probability that two distinct verticesxandy lie on a three- cycle inG? Answ ...
18 Random Variables Thus far, we have focused on probabilities of events. For example, we computed the probability that you win ...
Chapter 18 Random Variables740 Similarly,Mis a function mapping each outcome another way: M.HHH/ D 1 M.THH/ D 0 M.HHT/ D 0 M.THT ...
18.2. Independence 741 LikewiseŒMD1çis the eventfT T T;HHHgand has probability1=4. More generally, any assertion about the value ...
Chapter 18 Random Variables742 ThenH 1 is independent ofM, since PrŒMD1çD1=4DPr MD 1 jH 1 D 1 DPr MD 1 jH 1 D 0 PrŒMD0çD ...
18.3. Distribution Functions 743 x, and the closely related cumulative distribution function, CDFR.x/, measures the probability ...
Chapter 18 Random Variables744 3= 36 6= 36 x 2 V 2 3 4 5 6 7 8 9 10 11 12 PDFT.x/ Figure 18.1 The probability density function f ...
18.3. Distribution Functions 745 Both PDFRand CDFRcapture the same information aboutR, so take your choice. The key point here i ...
Chapter 18 Random Variables746 If the elements ofVin increasing order area 1 ;a 2 ;:::;an, then the cumulative distribution func ...
18.3. Distribution Functions 747 There is some probability that you guess correctly. In this case, you win 100% of the time. On ...
Chapter 18 Random Variables748 choices of x number peeked at TDH TDL TDH TDL TDH TDL 1=2 1=2 1=2 1=2 1=2 1=2 L=n .H�L/=n .n�H/=n ...
18.3. Distribution Functions 749 Randomized Algorithms The best strategy to win the numbers game is an example of arandomized al ...
Chapter 18 Random Variables750 f 20 .k/ 0:18 0:16 0:14 0:12 0:10 0:08 0:06 0:04 0:02 0 k 0 5 10 15 20 Figure 18.4 The pdf for th ...
18.4. Great Expectations 751 f20;:75.k/ 0:25 0:2 0:15 0:1 0:05 0 k 0 5 10 15 20 Figure 18.5 The pdf for the general binomial dis ...
Chapter 18 Random Variables752 18.4.1 The Expected Value of a Uniform Random Variable Rolling a 6-sided die provides an example ...
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