Suppose that X k is a time-homogenous Markov chain. Show thatP{X3= j3, X2= j2|X0= j0,X1 = j1}= P{X3 = j3 | X2 = j2} P{X2 = j2|X1 = j1}.
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Suppose that X k is a time-homogenous Markov chain. Show that
P{X3= j3, X2= j2|X0= j0,X1 = j1}= P{X3 = j3 | X2 = j2} P{X2 = j2|X1 = j1}.
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- Explain how you can determine the steady state matrix X of an absorbing Markov chain by inspection.A consultant's salary, captured by the random variable Y = B + X comes from a deterministic base B = 78 and a random bonus X. The bonus has mean E[X] = 16 and variance V[X] =240. What is the expected value of the total compensation E[Y]?Consider two machines that are maintained by a single repairman. Each machine functions for an exponentially distributed amount of time with rate λ=2 before it fails. The repair times for each unit are exponential with rate μ=1.The Markov chain model for this situation has state space indicating the number of machines that are in the repair shop: S={0,1,2}. Notice that you move from 0 to 1 if one of the two machines breaks down. You move from 1 to 0 when the machine in the repair room is repaired.
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- why is the covariance of a deterministic and a stochastic process 0? This relats to Arithmetic Bronian MotionConsider two machines that are maintained by a single repairman. Each machine functions for an exponentially distributed amount of time with rate λ before it fails. The repair times for each unit are exponential with rate μ. Formulate a Markov chain model for this situation with state space indicating the number of machines that are in the repair shop: S={0,1,2}. Notice that you move from 0 to 1 if one of the two machines breaks down. You move from 1 to 0 when the machine in the repair room is repaired. 2. Same as above but now machines are repaired in the order in which they fail. Each machine functions for an exponentially distributed amount of time with rate λi before it fails. The repair times for each unit are exponential with rate μi. The state space has nodes that are keep track of the machine that is at the repair shop (in case there is only one) and keeps track of which machine is worked on in case there are two machines at the repair shop. That is the state space is S={0,…Every day, Eric takes the same street from his home to the university. There are 4 streetlights along his way, and Eric has noticed the following Markov dependence. If he sees agreen light at an intersection, then 60% of time the next light is also green, and 40% of timethe next light is red. However, if he sees a red light, then 70% of time the next light is alsored, and 30% of time the next light is green.(a) Construct the transition probability matrix for the street lights.(b) If the first light is green, what is the probability that the third light is red?(c) Eric’s classmate Jacob has many street lights between his home and the university. Ifthe first street light is green, what is the probability that the last street light is red?(Use the steady-state distribution.)
- Let X1, X2, X3 be random variables such that Var(X1) = 5, Var(X2) = 4, Var(X3) = 7, cov(X1, X2) = 3, cov(X1, X3) = -2 and X2 and X3 are independent. Find the covariance between Y1 = X1 – 2X2 + 3X3 and Y2 = -2X1 + 3X2 + 4X3. %3DSuppose that {Xn} is a Markov chain with state space S = {1, 2}, transition matrix （1/5 4/5 2/5 3/5）, and initial distribution P (X0 = 1) = 3/4 and P (X0 = 2) = 1/4. Compute the following: P(X2 =2)Show that a Markov chain with transition matrix 1 P = |1/4 1/2 1/4 0 1 has more than one stationary distributions. Find the matrix that P" converges to, as
