1. Let Z =={Zn: n = 0, 1, } be a Markov chain defined on a state space S {1, ,m}.Let the one-step transition probability matrix M of Z have zero diagonal element, i.e.,M(i, j) =0,qijqij=ij‡i,=for any state (i, j) € S with qij ≥ 0 and qiΣjti qij > 0. Let N = {N₁ t≥ 0} be atime-homogeneous Poisson process with intensity > 0 and No = 0. Assume that Zand N are independent.Consider a stochastic processX₁ZN₁, for t≥ 0.=(1)The (i, j)-element of transition matrix P(t) of X is defined by the conditional probabilitywhere the matrix Q is defined byCOP(X, = j|Xo = i) = Σ P(Zn = j, N, = n|Zo = i).n=0(a) Show that X is a continuous-time Markov chain with transition matrixP(t) = eºt,(2)Q = D(M-I),where I is a (m x m)-identity matrix and D is a diagonal matrix diag(q₁,..., 9m).Hint for a square matrix A, E = ¹.A"n!(b) Show that Q is an intensity matrix.

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A First Course in Probability (10th Edition)

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For the attached transition probability matrix for a Markov chain with {Xn ; n = 0, 1, 2,.........}: a) How many classes exist, and which two states are the absorption states? b) What is the limn->inf P{Xn = 3 | X0 = 3}? b) What is the limn->inf P{Xn = 1 | X0 = 3}?
A 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.]
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 1
9. Let i be a transient state of a continuous-time Markov chain X with X (0) : = i. Show that the total time spent in state i has an exponential distribution.
(6) Give an example of a Markov chain Xn on a countable state space S and a function g on S such that g(Xn) is not a Markov chain. Can you give any conditions on Xn and g which ensure that g(Xn) is a Markov chain?
Let (X0, X1, X2, . . .) be the discrete-time, homogeneous Markov chain on state space S = {1, 2, 3, 4, 5, 6} with X0 = 1 and transition matrix
IN Markov process having transition matrix A = [a,k], whose entries are a11 = a12 = 0.6,a21 = 0.8, a22 = 0.8. and the initial state [0.7 0.8]T, SOLVE FOR the next 3 states.
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22 Let X and Y be jointly Gaussian random variables where oy= oy and p =-1. Find a transformation matrix such that new random variables Y, and Y, are statistically independent.
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