/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 26 A flea moves around the vertices... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 26

A flea moves around the vertices of a triangle in the following manner: Whenever it is at vertex \(i\) it moves to its clockwise neighbor vertex with probability \(p_{i}\) and to the counterclockwise neighbor with probability \(q_{i}=1-p_{i}, i=1,2,3\) (a) Find the proportion of time that the flea is at each of the vertices. (b) How often does the flea make a counterclockwise move which is then followed by 5 consecutive clockwise moves?

Short Answer

Expert verified
The short answer for the given problem: (a) The proportion of time that the flea is at each of the vertices can be found by solving a system of linear equations to find the stationary distribution of the Markov chain. The stationary distribution is given by \((\pi_1, \pi_2, \pi_3)\), where \(\pi_i\) is the proportion of time spent at vertex \(i\). (b) The probability that the flea makes a counterclockwise move followed by 5 consecutive clockwise moves is given by: \[Prob = \sum_{i=1}^{3} \sum_{j=1}^{3} \pi_j A_{j,i}\] where \(A_{j,i} = q_j \cdot p_{j-1}^5\) (indices modulo 3, with 0 taken as 3). Calculate the probabilities \(A_{j,i}\) and then compute the probability of the specified sequence by summing over all possible starting vertices \(j\) and ending vertices \(i\).

Step by step solution

01

Markov Chain Representation

We can represent this problem as a Markov chain with 3 states: vertex 1, vertex 2, and vertex 3. The transition probabilities are given in the problem: for each vertex, the probability of moving clockwise is \(p_i\) and counterclockwise is \(q_i\). The transition matrix P can be written as: \[P = \begin{bmatrix} 0 & p_1 & q_1 \\ q_2 & 0 & p_2 \\ p_3 & q_3 & 0 \end{bmatrix}\]
02

(a) Finding Stationary Distribution

We want to find the stationary distribution, which is the row vector \(\pi = (\pi_1, \pi_2, \pi_3)\) such that: 1. \(\pi P = \pi\) 2. \(\sum_{i=1}^3 \pi_i = 1\) We can solve this by setting up a system of linear equations: 1. \(\pi_1 = p_2\pi_3 + q_1\pi_2\) 2. \(\pi_2 = p_3\pi_1 + q_2\pi_3\) 3. \(\pi_3 = p_1\pi_2 + q_3\pi_1\) 4. \(\pi_1 + \pi_2 + \pi_3 = 1\) We can solve this system of linear equations to find the stationary distribution.
03

(a) Solving Linear System

We can solve the system using any linear algebra method (e.g. substitution, elimination, or matrix inversion). Without loss of generality, we can eliminate \(\pi_3\) by expressing all variables in terms of \(\pi_1\) and \(\pi_2\): 1. \(\pi_3 = \frac{q_1\pi_2 - \pi_1}{p_2}\) 2. \(\pi_3 = \frac{p_3\pi_1 - \pi_2}{q_2}\) 3. \(\pi_3 = 1 - \pi_1 - \pi_2\) We can equate the first two expressions for \(\pi_3\) and then solve for \(\pi_1\) and \(\pi_2\) in terms of the unknowns \(p_i\) and \(q_i\). Finally, substitute these values of \(\pi_1\) and \(\pi_2\) into the third equation to get \(\pi_3\). After solving the system of linear equations, we will get the stationary distribution \((\pi_1, \pi_2, \pi_3)\). This represents the proportion of time that the flea is at each of the vertices.
04

(b) Probability of the Specified Sequence

We want to find the probability that the flea makes a counterclockwise move (say, to vertex \(i\)), followed by 5 consecutive clockwise moves. Let \(A_{j,i}\) denote the probability that starting at vertex \(j\), the flea makes a CCW move to vertex \(i\), followed by 5 consecutive CW moves. Then the probability of the specified sequence is given by: \[Prob = \sum_{i=1}^{3} \sum_{j=1}^{3} \pi_j A_{j,i}\] Now, \(A_{j,i} = q_j \cdot p_{j-1}^5\) where indices are considered modulo 3 (i.e., 0 taken as 3). Thus, probabilities \(A_{j,i}\) can be calculated, and the probability of the specified sequence can be found by summing over all possible starting vertices \(j\) and ending vertices \(i\).

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Most popular questions from this chapter

Let the transition probability matrix of a two-state Markov chain be given, as in Example 4.2, by $$ \mathbf{P}=\left|\begin{array}{cc} p & \mid-p \\ 1-p & p \end{array}\right| $$ Show by mathematical induction that $$ \mathbf{P}^{(n)}=\left|\begin{array}{ll} \frac{1}{2}+\frac{1}{2}(2 p-1)^{n} & \frac{1}{2}-\frac{1}{2}(2 p-1)^{n} \\ \frac{1}{2}-\frac{1}{2}(2 p-1)^{n} & \frac{1}{2}+\frac{1}{2}(2 p-1)^{n} \end{array}\right| $$

Suppose that on each play of the game a gambler either wins 1 with probability \(p\) or loses 1 with probability \(1-p\). The gambler continues betting until she or he is either winning \(n\) or losing \(m\). What is the probability that the gambler quits a winner?

16\. Let \(Y_{n}\) be the sum of \(n\) independent rolls of a fair dic. Find $$ \lim _{n \rightarrow \infty} P\left[Y_{n} \text { is a multiple of } 13\right] $$ Define an appropriate Markov chain and apply the results of Exercise 14.

Specify the classes of the following Markov chains, and determine whether they are transient or recurrent: $$ \begin{aligned} &\mathbf{P}_{1}=\left|\begin{array}{ccc} 0 & \frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & 0 & \frac{1}{2} \\ \frac{1}{2} & \frac{1}{2} & 0 \end{array}\right| & \mathbf{P}_{2}=\left|\begin{array}{cccc} 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 \\ 1 & \frac{1}{2} & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array}\right| \\ &\mathbf{P}_{3}=\left\|\begin{array}{lllll} \frac{1}{2} & 0 & \frac{1}{2} & 0 & 0 \\ \frac{1}{4} & \frac{1}{2} & \frac{1}{4} & 0 & 0 \\ \frac{1}{2} & 0 & \frac{1}{2} & 0 & 0 \\ 0 & 0 & 0 & \frac{1}{2} & \frac{1}{2} \\ 0 & 0 & 0 & \frac{1}{2} & \frac{1}{2} \end{array}\right\| & \mathbf{P}_{4}=\mid \begin{array}{lllll} \frac{1}{2} & \frac{1}{4} & 0 & 0 & 0 \\ \frac{1}{2} & \frac{1}{2} & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & \vdots & 3 & 0 \\ 1 & 0 & 0 & 0 & 0 \end{array} \| \end{aligned} $$

An individual possesses \(r\) umbrellas which he employs in going from his home to office, and vice versa. If he is at home (the office) at the beginning (end) of a day and it is raining, then he will take an umbrella with him to the office (home), provided there is one to be taken. If it is not raining, then he never takes an umbrella. Assume that, independent of the past, it rains at the beginning (end) of a day with probability \(p\). (i) Define a Markov chain with \(r+1\) states which will help us to determine the proportion of time that our man gets wet. (Note: He gets wet if it is raining, and all umbrellas are at his other location.) (ii) Show that the limiting probabilities are given by $$ \pi_{i}=\left\\{\begin{array}{ll} \frac{q}{r+q}, & \text { if } i=0 \\ \frac{1}{r+q}, & \text { if } i=1, \ldots, r \end{array} \quad \text { where } q=1-p\right. $$ (iii) What fraction of time does our man get wet? (iv) When \(r=3\), what value of \(p\) maximizes the fraction of time he gets wet?
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