A Markov chain has transition matrix 글 0 글 3 Given the initial probabilities ø1 = $2 = $3 = , find Pr (X1 # X2). %3D
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- 5. Let P be the transition matrix of a Markov chain with finite state space. Let I be the identity matrix, U the |S| x |S| matrix with all entries unity, and 1 the row |S|-vector with all entries unity. Let л be a non-negative vector with Σ; i = 1. Show that лP = if and only if (I-P+U) = 1. Deduce that if P is irreducible then = 1(I-P + U)-¹.11. A certain mobile phone app is becoming popular in a large population. Every week 10% of those who are not using the app, either because they don't have it yet or have it but are not using it, start using it, and 15% of those who are using it stop using it. Assume that the starting percentages are that 80% are not using it and 20% are using it. (a) Show the Markov matrix A representing the situation. (Letting .80 represent the starting 1.20 .80 percentages, remember that you want the situation after one week to be given by A .) .20 (b) What percentage will be using the app after two weeks? (c) Find the eigenvalues and eigenvectors of A. (d) Show a matrix X which diagonalizes A by means of X-'AX. (d) In the long run the percentages not using the app will be Show your work. _% and using it will be _%7
- Can someone please help me with this question. I am having so much trouble.Give me right solution according to the questionA study of armed robbers yielded the approximate transition probability matrix shown below. The matrix gives the probability that a robber currents free, on probation, or in jail would, over a period of a year, make a transition to one of the states. То From Free Probation Jail Free 0.7 0.2 0.1 Probation 0.3 0.5 0.2 Jail 0.0 0.1 0.9 Assuming that transitions are recorded at the end of each one-year period: i) For a robber who is now free, what is the expected number of years before going to jail? ii) What proportion of time can a robber expect to spend in jail? [Note: You may consider maximum four transitions as equivalent to that of steady state if you like.]
- Find an optimal parenthesisation of a matrix chain product whose sequence of dimensions is(4, 8, 7, 2, 3).Consider a continuous-time Markov chain whose jump chain is a random walk with reflecting barriers 0 and m where po,1 = 1 and pm,m-1 =1 and pii-1 = Pii+1 = for 1If A is a Markov matrix, why doesn't I+ A+ A2 + · · · add up to (I -A)-1?Consider a Markov chain with two possible states, S = {0, 1}. In particular, suppose that the transition matrix is given by Show that pn = 1 P = [¹3 В -x [/³² x] + B x +B[B x] x 1- B] (1-x-B)¹ x +B x -|- -B ·X₁ ВConsider a random walk on the graph with 6 vertices below. Suppose that 0 and 5 are absorbing vertices, but the random walker is more strongly attracted to 5 than to 0. So for each turn, the probability that the walker moves right is 0.7, while the probability he moves left is only 0.3. (a) Write the transition matrix P for this Markov process. (b) Find POO. (c) What is the probability that a walker starting at vertex 1 is absorbed by vertex 0? (d) What is the probability that a walker starting at vertex 1 is absorbed by vertex 5? (e) What is the probability that a walker starting at vertex 4 is absorbed by vertex 5? (f) What is the probability that a walker starting at vertex 3 is absorbed by vertex 5? What is the expected number of times that a walker starting at vertex 1 will visit vertex 2? (h) What is the expected number of times that a walker starting at vertex 1 will visit vertex 4? N;B The diagram is a horizontal line showing points 0, 1, 2, 3, 4, and 55SEE MORE QUESTIONS
