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)

10th Edition

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Chapter1: Combinatorial Analysis

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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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4. Let X be a Markov chain with state space S = {1, 2, 3) and transition matrix (² where 0 < p < 1. Prove that P = 0 Ph = P P P 1-P 0 0 P 1-p ain a2n a3n aln a2n a3n where ain + wa2n + w²a3n = (1 − p + pw)", w being a complex cube root of 1. a3n a2n ain
Q. No. 42 Let {X,}nzo be a homogeneous Markov chain whose state space is {0,1,2} and 1 whose one-step transition probability matrix is P = 0.3 0.7. Then 1 lim P(X2n = 2 |X, = 2) = (correct up to one decimal place). n-00
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.]
A Markov chain {s} with state space n={1, 2, 3} has a sequence of realizations and process transitions as follows: Transition (1,1) (1,2) (1,3) (2,1) (2,2) (2,3) (3.1) (3,2) (3,3) t S: S+1 2 1 1. 1 1 1 1 1 3 1 3 3 3 4 3 1. 2 3 1. 2 1 7 2 2 1. 8 2 2 1 9 2 3 1. 10 3 11 1 1 12 1 1. 13 1 1. 14 1. 1. 15 3 1. 1. 16 1. Number of frequencies : 1 3. 2 2 1 1 1 1 a. determine the transition frequency matrix O= 2 12 93 021 022 °23 %3D 31 °32 °33) 1 P1 P12 P13 b. determine the estimated transition probability matrixP= 2 P21 P22 P23 3 P31 P32 P33)
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
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Which of the following transition matrices is/are for a regular Markov Chain? 0 0 1 X = % 0 0 0 1 Y =|0 ½ ½ Z = % 0 2 (A) Y (В) X, Y (С) х (D) Y, Z (E) none (F) X, Z (G) (Н) Х, Y, Z
Please answer #28 and explain your reasoning.
1. Prove that Weiner Process is Markovian (Markov Process).
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.
A path of length k in a Markov chain {X,,n= 0,1,...} is a sequence of states visited from step n to step n+k. Let pij be the transition probability from state i to state j. Show that starting from state i,, the probability that the chain follows a particular path i, - i„.1 - in2 - ... - i,t is given by %3D "x*\":= "X)d = i2.,Xpsk = inrk | X, = i,) n+1 n+l> n+2 n+k = P Pu.

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