Let X(n), n 0 be the two-state Markov chain on states (0,1) with transition probability matrix...
(n)," 2 0) be the two-state Markov chain on states (. i} with transition probability matrix 0.7 0.3 0.4 0.6 Find P(X(2) 0 and X(5) X() 0)
Let p E [0,1] with pメ, and let (Xn)n=o b l e the Markov chain on with initia [0,1] given by distribution δο and transition matrix 11: Z Z ify=x-1 p 0 otherwise. Use the strong law of large numbers to show that each state is transient. Hint: consider another Markov chain with additional structure but with the same distribution and transition matrix
Let p E [0,1] with pメ, and let (Xn)n=o b l e the Markov chain on with...
6. (6 poiats) Let (X(n),n 2 0) be the two-state Markow chain on states (o,I with transition 1-6 61 probability matrix Find: (a) P(x(1)-oX(0) ,x(2) 0) Note. (b) is an unconditional joint robability so you will need to include the initial distribution P(X(0) = 0)-π。(0) and P(X(0) = 1)- (0)
6. (6 poiats) Let (X(n),n 2 0) be the two-state Markow chain on states (o,I with transition 1-6 61 probability matrix Find: (a) P(x(1)-oX(0) ,x(2) 0) Note. (b) is an unconditional joint robability so you will need to include the initial distribution P(X(0) = 0)-π。(0) and P(X(0) = 1)- (0)
Consider the Markov chain X0,X1,X2,... on the state space S = {0,1} with transition matrix P= (a) Show that the process defined by the pair Zn := (Xn−1,Xn), n ≥ 1, is a Markov chain on the state space consisting of four (pair) states: (0,0),(0,1),(1,0),(1,1). (b) Determine the transition probability matrix for the process Zn, n ≥ 1.
Let Xn be a Markov chain with state space {0,1,2}, the initial
probability vector and one step transition matrix
a. Compute.
b. Compute.
3. Let X be a Markov chain with state space {0,1,2}, the initial probability vector - and one step transition matrix pt 0 Compute P-1, X, = 0, x, - 2), P(X, = 0) b. Compute P( -1| X, = 2), P(X, = 0 | X, = 1) _ a.
3. Let X be a Markov chain...
Consider a two state Markov chain with one-step transition matrix on the states 1,21, , 0<p+q<2. 91-9 ' Show, by induction or otherwise, that the n-step transition matrix is Ptg -99 Based upon the above equation, what is lim-x P(Xn-2K-1). How about limn→x P(Xn-
Let Xn be a Markov chain with state space {0, 1, 2}, and transition probability matrix and initial distribution π = (0.2, 0.5, 0.3). Calculate P(X1 = 2) and P(X3 = 2|X0 = 0) 0.3 0.1 0.6 p0.4 0.4 0.2 0.1 0.7 0.2
P is the (one-step) transition probability matrix of a Markov chain with state space {0, 1, 2, 3, 4 0.5 0.0 0.5 0.0 0.0 0.25 0.5 0.25 0.0 0.0 P=10.5 0.0 0.5 0.0 0.0 0.0 0.0 0.0 0.5 0.5 0.0 0.0 0.0 0.5 0.5/ (a) Draw a transition diagram. (b) Suppose the chain starts at time 0 in state 2. That is, Xo 2. Find E Xi (c)Suppose the chain starts at time 0 in any of the states with...