Mathematics for Computer Science
19.7. Really Great Expectations 833 here are probabilities that I forget various pieces of footwear: left sock 0:2 right sock 0: ...
Chapter 19 Deviation from the Mean834 tranches of 100 bonds each. Then, all the defaults must fill up the lowest tranche before ...
19.7. Really Great Expectations 835 Problem 19.32. An infinite version of Murphy’s Law is that if an infinite number of mutually ...
Chapter 19 Deviation from the Mean836 (b)Prove that VarŒRçis infinite. A joking way to phrase the point of this example is “the ...
19.7. Really Great Expectations 837 A gambler bets $10 on “red” at a roulette table (the odds of red are 18/38, slightly less th ...
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20 Random Walks Random Walksare used to model situations in which an object moves in a se- quence of steps in randomly chosen di ...
Chapter 20 Random Walks840 capital gambler’s n T = n + m time bet outcomes: WLLWLWWLLL Figure 20.1 A graph of the gambler’s capi ...
20.1. Gambler’s Ruin 841 every time a Head appears, Albert wins $1 from Eric, and vice versa for Tails. They play this game unti ...
Chapter 20 Random Walks842 same reasoning that every bet has fair worth. So, Albert’s expected worth at the end of the game is t ...
20.1. Gambler’s Ruin 843 20.1.2 A Recurrence for the Probability of Winning Fortunately, you don’t need to be as ingenuious Pasc ...
Chapter 20 Random Walks844 for some constantsA;B. To solve forA;B, note that by (20.6) and (20.7), w 1 xDA.1rx/CB.1x/; so lettin ...
20.1. Gambler’s Ruin 845 Corollary 20.1.2.In the Gambler’s Ruin game with initial capital,n, target,T, and probabilityp < 1=2 ...
Chapter 20 Random Walks846 w n 0 downward drift gambler’s wealth time upward swing (too late) Figure 20.2 In a biased random wal ...
20.1. Gambler’s Ruin 847 For fixedpandT, letenbe the expected number of bets until the game ends when the gambler’s initial capi ...
Chapter 20 Random Walks848 20.1.5 Quit While You Are Ahead Suppose that the gambler never quits while he is ahead. That is, he s ...
20.2. Random Walks on Graphs 849 1 10 ^100 , that you will go broke immediately. But there is a 10 ^100 probability that you wi ...
Chapter 20 Random Walks850 to a typical search. For example, on May 2, 2012, a search on Google for “ ‘Mathe- matics for Compute ...
20.2. Random Walks on Graphs 851 There is another problem—a page could become unfairly influential by having lots of links to ot ...
Chapter 20 Random Walks852 For example, in our web graph, we have PrŒgo tox 4 çD PrŒatx 7 ç 2 C PrŒatx 2 ç 1 : One can think of ...
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