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...

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5. (i) Let 1 be a permutation on {1,2, ...,n}. The matrix P for which pi. = em(i). is called the permutation matrix associated with 7, where Pie is the i-th row of Pr and e;j = 1 if i = j and 0 otherwise. Let n = 4 and T = (3 2 4 1). Find Pr. (ii) Find the corresponding permutation matrix for the permutation %3D 1 2 3 4 3 4 1 2
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?
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For a parallel structure of identical components, the system can succeed if at least one of the components succeeds, Assume that components fail independently of each other and that each component has a 0.08 probability of failure. Complete parts (a) through (c) below. (a) Would it be unusual to observe one component fail? Two components? It V be unusual to observe one component fail, since the probability that one component fails,, is V than 0.05. It V be unusual to observe two components fail, since the probability that two components fail,. is v than 0.05. (Type integers or decimals. Do not round.) (b) What is the probability that a parallel structure with 2 identical components will succeed? (Round to four decimal places as needed.) (c) How many components would be needed in the structure so that the probability the system will succeed is greater than 0.9998? (Type a whole number.) P Type here to search lyp
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A B D 10. (a) Show the graph above and its adjacency matrix. (b) Find the number of walks of length 4 that start at B and end at A or C. Explain your work.
In at tropical forest, the populations of falcons and mice are related toeach other. The mice eat everything they can while the falcons eat the mice. The population sizes of falcons and mice evolve according to the rule. * At the beginning of 2023, there were 10 falcons and 40 mice. a) Approximately how many falcons and mice will there be at the beginning of 2068? b) c) Find a diagonalization, A = BDB-1, of the transition matrix A from part (b), where D is a 2x2 diagonal matrix, and B is an invertible2x2 matrix. d) Use your answer to (c) to estimate the (approximate) numbersof falcons and mice there will be at the beginning of 2058. e) what restrictions on the initial population sizes ensures the long time survival of both species in the forest
A market demand has 3 possible states namely GOOD, NORMAL and BAD for each period. At each period there are 2 possible decisions for the manager as do nothing/ make promotion. The transition matrix of states regarding for possible decisions are given as below Pof do nothing GOOD NORMAL BAD GOOD 0.4 0.2 04 0.3 0.2 0.5 NORMAL 0.5 BAD 0.1 0.4 Pof make promotion GOOD NORMAL BAD 0.4 0.4 0.5 GOOD 0.2 NORMAL 04 0.1 BAD 0.3 0.4 0.3 It is known that the income for each period when the state in GOOD, NORMAL and BAD are 10000, 7000, 2000. Cost for do nothing is 0, and make promotion is 3000. a) Given at period 0 the state is NORMAL, estimate expected beneft obtain two periods when the sequence of decisions is do nothing, do nothing b) Given at period 0 the state is NORMAL, estimate expected beneft obtain two periods when the sequence of decisions is make promotion, make promotion
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