A rat is put into the following maze: The rat has a probability of 1/4 of starting in any compartment and suppose that the rat chooses a passageway at random when it makes a move from one compartment to another at each time. Let Xn be the compartment occupied by the rat after n moves. Explain why {Xn} is a MC and find the transition matrix P . Explain why the chain is irreducible, aperiodic and positive What is the limit of Pn? Find the probability that the rat is in compartment 3 after two In the long run, how many times that the rat enters in compartment 4 in 100 mov

A rat is put into the following maze: The rat has a probability of 1/4 of starting in any compartment and suppose that the rat chooses a passageway at random when it makes a move from one compartment to another at each time. Let Xn be the compartment occupied by the rat after n moves. Explain why {Xn} is a MC and find the transition matrix P . Explain why the chain is irreducible, aperiodic and positive What is the limit of Pn? Find the probability that the rat is in compartment 3 after two In the long run, how many times that the rat enters in compartment 4 in 100 mov

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

See similar textbooks

Related questions

Q: Let G be the graph below. E D B C (a) Construct the transition matrix for a random walk around G.

A: The given graph G is as follows The transition matrix P is a representation of a random walk. P…

Q: Which matrix represents the number of walks of length 2 and how many walks of length 2 are in the…

Q: Staff at the fast food outlet start out as trainees (Tr). Trainees either go on to become…

A: We are alllow to solve 4 subpart of a question If yoou need rest of answere please upload it…

Q: Out(4= G7. Present the adjacency matrix for the graph above:

Q: Please do the questions with handwritten working. I'm struggling to understand what to write

Q: Show by example that chains which are not irreducible may have many different stationary…

Q: 4. Let A be a square matrix with transpose A". • If AT = A, then A is called a symmetric matrix. %3D…

A: If A is non singular then A is non zero. ATT = A

Q: Suppose a math professor collects data on the probability that students attending a given class…

Q: -23 16 24 Let A = -6 5 . Find an invertible matrix P and a diagonal matrix D such that D = P-'AP.…

Q: Suppose A is a matrix. Let B be given by switching two of the rows of A. What is the relation…

A: Switching two rows of a matrix is equivalent to multiplying the matrix by a permutation matrix that…

Q: Given the adjacency relation p = {(2, 1), (3, 2), (3, 3), (3, 4), (4, 5), (6, 3), (6, 6)} on the set…

A: According to the given information, the given adjacency relation is: It is required to find the…

Q: Consider an unfair coin with probability p of heads. n has been tossed (a deterministic number) n…

A: Note: Hi there! Thank you for posting the question. As you have posted multiple questions, as per…

Q: Two subjects, Pure Mathematies and Applied Mathematics, are offered in the first semester of a…

A: In the question we have to find matrixes. NOTE - As per BARTLEBY's policy we can not solve…

Q: Assume B is an n × n matrix, n > 1, and det(B) = 4. (a) Find n such that det(−5B) = 100

A: Given:We are required to find:

Q: Prove that the product of two n x n anti-diagonal matrices is diagonal.

A: To prove that the product of two n*n anti-diagonal matrices is a diagonal matrix.

Q: 11. Let P= Va be the transition matrix for a Markov chain. In the long-run, what is the probability…

A: Given: P=012/31/3 Substitute respective values in the equation Pπ=π. 012/31/3π1π2=π1π2

Q: We want to train a mouse to follow a particular path in the maze (see image 1). To measure the…

A: (a) The objective is to construct the matrix T describing the behavior of the untrained mouse, where…

Q: ₁ = [ 11, 11] Find a nonsingular matrix P such that PAP is diagonal where A = -5 P = = [00] OP - [3…

Q: What is the transition matrix from S, to S, given

Question

A rat is put into the following maze:

The rat has a probability of 1/4 of starting in any compartment and suppose that the rat chooses a passageway at random when it makes a move from one compartment to another at each time. Let X_n be the compartment occupied by the rat after n moves.

Explain why {X_n} is a MC and find the transition matrix P .
Explain why the chain is irreducible, aperiodic and positive
What is the limit of Pⁿ?
Find the probability that the rat is in compartment 3 after two
In the long run, how many times that the rat enters in compartment 4 in 100 movements?

Video Video

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

SEE SOLUTION Check out a sample Q&A here

Step 1

Step 2

Step 3

Step 4

Step by step

Solved in 4 steps with 1 images

SEE SOLUTION Check out a sample Q&A here

Similar questions

2. Let G be the following graph. V1 V2 V3 V5 VA (a) Let A be the adjacency matrix of G. Find A. (b) Using only A and its powers, determine how many walks of length EXACTLY 3 are there starting at 2 and ending at v4. Explain. (c) Using only A and its powers, determine how many walks of length AT MOST 3 are there starting at 2 and ending at v4. Explain.
3. A rat is put into the following maze: 3 moves. 1 2 The rat has a probability of 1/4 of starting in any compartment and suppose that the rat chooses a passageway at random when it makes a move from one compartment to another at each time. Let Xn be the compartment occupied by the rat after n (a) Explain why {X₂} is a MC and find the transition matrix P. (b) Explain why the chain is irreducible, aperiodic and positive recurrent. (c) What is the limit of Pn? Explain. (d) Find the probability that the rat is in compartment 3 after two moves. (e) In the long run, how many times that the rat enters in compartment 4 in 100 movements?
Help me fast so that I will give Upvote.
We want to train a mouse to follow a particular path in the maze (see image 1). To measure the effectiveness of the training, we want to compare the behaviour of the trained mouse to that of a mouse that runs through the maze at random.(a) Construct the matrix T describing the behaviour of the untrained mouse, where tij gives the probability of going to room i when the mouse is in room j. (Note that since the mouse moves randomly if it is in front of two doors, it has a 1/2 probability of going through each of these doors.) (b) Suppose the mouse begins its journey through the maze with a vector of probabilities P(0) = (see image 3) where pi(0) represents the probability of being in room i at the start; so of course we have p1(0)+p2(0)+p3(0)= 1). Show that the probabilities of being in room i after a change of room P(1) are given by P(1)=TP(0). Indications: proceed with a proof on the one hand and on the other hand: start by calculating the probabilities of being in parts 1,2 or 3 after…
4. Let M be a real square matrix such that M3 + M = 2M². Prove that M is diagonalisable. 5. Connect to you cocalc account, read the handout and make the assignment there. It will be collected automatically on 29 March 2022 and counts as one of the exercises of this exercise sheet.
3. suppose that the manufacturer keeps a spare machine that only is used when the primary machine is being repaired. During a repair day, the spare machine has a probability of 0.1 of breaking down, in which case it is repaired the next day. Denote the state of the system by (x, y), where x and y, respectively, take on the values 1 or 0 depending upon whether the primary machine (x) and the spare machine (y) are operational (value of 1) or not operational (value of 0) at the end of the day. [Hint: Note that (0, 0) is not a possible state.] (a) Construct the (one-step) transition matrix for this Markov chain. (b) Find the expected recurrence time for the state (1, 0).
Suppose in a scatterplot matrix, you observe that all of the scatterplots associated with the explanatory variables show a strong linear relationship. There are three explanatory variables in the model. Of the following, which is the most valid to do? Remove the response variable. Remove only one of the explanatory variables and keep only two. Remove all of the explanatory variables because they are linearly related to each other and therefore explain the same thing. Remove exactly two of the explanatory variables because they are all linearly related to each other and therefore explain the same thing. We only need to keep one in the model.
8. Let A be an m x n matrix. Prove that if n is an odd number, then rank(A) # nullity(A).
M P₁ 0000 P3 Give the adjacency matrix M associated with the directed graph above. P5 ОРА Those longest possible walks all start at the vertex P1 for the vertex P₁, P2 for the vertex P₂, etc. P₂ The smallest integer value n for which M" = 0 (where 0 represents a zero matrix) is This indicates that the longest walk between any two vertices in the graph would have length and end at the vertex (based on the matrix M"-1). Write
2. Find the adjacency matrix for the following graph. d G7