7.40. A random walk on a graph goes from vertex to adjacent vertex by walking along the connecting edge. It chooses the edge at random: all edges connecting to that vertex are equally likely. Consider a random walk on the graph below. (a) Find the long-term proportion of time spent at vertex 0. (b) Find the expected return time to 0. (c) Find the probability that a random walk from vertex e hits 0 before hitting either a or b. (d) Suppose the random walk starts at 0. Find the expected time to reach {a Ub}. [Hint: Make use of symmetry!] A
7.40. A random walk on a graph goes from vertex to adjacent vertex by walking along the connecting edge. It chooses the edge at random: all edges connecting to that vertex are equally likely. Consider a random walk on the graph below. (a) Find the long-term proportion of time spent at vertex 0. (b) Find the expected return time to 0. (c) Find the probability that a random walk from vertex e hits 0 before hitting either a or b. (d) Suppose the random walk starts at 0. Find the expected time to reach {a Ub}. [Hint: Make use of symmetry!] A
Chapter8: Sequences, Series,and Probability
Section8.7: Probability
Problem 11ECP: A manufacturer has determined that a machine averages one faulty unit for every 500 it produces....
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![7.40. A random walk on a graph goes from vertex to adjacent vertex by walking
along the connecting edge. It chooses the edge at random: all edges connecting to
that vertex are equally likely. Consider a random walk on the graph below.
(a) Find the long-term proportion of time spent at vertex 0.
(b) Find the expected return time to 0.
(c) Find the probability that a random walk from vertex e hits 0 before hitting
either a or b.
(d) Suppose the random walk starts at 0. Find the expected time to reach
{a Ub}.
[Hint: Make use of symmetry!]
A](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe15ed467-90ec-4e60-afef-3d3f6119f74d%2F8a641b14-37a8-4ec0-a50e-fc992b9a7fe5%2F3qvmh2s_processed.png&w=3840&q=75)
Transcribed Image Text:7.40. A random walk on a graph goes from vertex to adjacent vertex by walking
along the connecting edge. It chooses the edge at random: all edges connecting to
that vertex are equally likely. Consider a random walk on the graph below.
(a) Find the long-term proportion of time spent at vertex 0.
(b) Find the expected return time to 0.
(c) Find the probability that a random walk from vertex e hits 0 before hitting
either a or b.
(d) Suppose the random walk starts at 0. Find the expected time to reach
{a Ub}.
[Hint: Make use of symmetry!]
A
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