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Mathematics for Computer Science
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“mcs” — 2017/3/3 — 11:21 — page 923 — #931<br />
21.2. Random Walks on Graphs 923<br />
Problems <strong>for</strong> Section 21.2<br />
Practice Problems<br />
Problem 21.5.<br />
Consider the following random-walk graphs:<br />
1<br />
x<br />
y<br />
1<br />
Figure 21.4<br />
w<br />
1<br />
0.9<br />
z<br />
0.1<br />
Figure 21.5<br />
1/2<br />
1/2<br />
1 a b c<br />
1/2<br />
d<br />
1<br />
1/2<br />
Figure 21.6<br />
(a) Find d.x/ <strong>for</strong> a stationary distribution <strong>for</strong> graph 21.4.<br />
(b) Find d.y/ <strong>for</strong> a stationary distribution <strong>for</strong> graph 21.4.<br />
(c) If you start at node x in graph 21.4 and take a (long) random walk, does the<br />
distribution over nodes ever get close to the stationary distribution?<br />
(d) Find d.w/ <strong>for</strong> a stationary distribution <strong>for</strong> graph 21.5.
“mcs” — 2017/3/3 — 11:21 — page 922 — #930 922 Chapter 21 Random Walks Conclude that starting with a 1 dollar stake in a fair game, the gambler can expect to play <strong>for</strong>ever! Problem 21.3. A gambler is placing $1 bets on the “1st dozen” in roulette. This bet wins when a number from one to twelve comes in, and then the gambler gets his $1 back plus $2 more. Recall that there are 38 numbers on the roulette wheel. The gambler’s initial stake in $n and his target is $T . He will keep betting until he runs out of money (“goes broke”) or reaches his target. Let w n be the probability of the gambler winning, that is, reaching target $T be<strong>for</strong>e going broke. (a) Write a linear recurrence with boundary conditions <strong>for</strong> w n . You need not solve the recurrence. (b) Let e n be the expected number of bets until the game ends. Write a linear recurrence with boundary conditions <strong>for</strong> e n . You need not solve the recurrence. Problem 21.4. In the fair Gambler’s Ruin game with initial stake of n dollars and target of T dollars, let e n be the number of $1 bets the gambler makes until the game ends (because he reaches his target or goes broke). (a) Describe constants a; b; c such that <strong>for</strong> 1 < n < T . e n D ae n 1 C be n 2 C c: (21.17) (b) Let e n be defined by (21.17) <strong>for</strong> all n > 1, where e 0 D 0 and e 1 D d <strong>for</strong> some constant d. Derive a closed <strong>for</strong>m (involving d) <strong>for</strong> the generating function E.x/ WWD P 1 0 e nx n . (c) Find a closed <strong>for</strong>m (involving d) <strong>for</strong> e n . (d) Use part (c) to solve <strong>for</strong> d. (e) Prove that e n D n.T n/.
“mcs” — 2017/3/3 — 11:21 — page 923 — #931 21.2. Random Walks on Graphs 923 Problems <strong>for</strong> Section 21.2 Practice Problems Problem 21.5. Consider the following random-walk graphs: 1 x y 1 Figure 21.4 w 1 0.9 z 0.1 Figure 21.5 1/2 1/2 1 a b c 1/2 d 1 1/2 Figure 21.6 (a) Find d.x/ <strong>for</strong> a stationary distribution <strong>for</strong> graph 21.4. (b) Find d.y/ <strong>for</strong> a stationary distribution <strong>for</strong> graph 21.4. (c) If you start at node x in graph 21.4 and take a (long) random walk, does the distribution over nodes ever get close to the stationary distribution? (d) Find d.w/ <strong>for</strong> a stationary distribution <strong>for</strong> graph 21.5.
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