Please give the detail solution to the problems.
Please give the detail solution to the problems. Let (T,P) be a time-homogeneous discrete-time Markov chain...
1. Let (т, P) be a time-homogeneous discrete-time Markov chain with state space {1, . . . , (a) Show that the Markov chain is not stationary (i.e., SSS). (b) Suppose P is doubly stochastic and π- JJ, . . . , Đ. Then show that the Markov chain is stationary
1. Exit times. Let X be a discrete-time Markov chain (with discrete state space) and suppose pii > 0. Let T =min{n 21: X i} be the exit time from state i. Show that T has a geometric distribution with respect to the conditional probability P 1. Exit times. Let X be a discrete-time Markov chain (with discrete state space) and suppose pii > 0. Let T =min{n 21: X i} be the exit time from state i. Show that...
2. A Markov chain is said to be doubly stochastic if both the rows and columns of the transition matrix sum to 1. Assume that the state space is {0, 1,....m}, and that the Markov chain is doubly stochastic and irreducible. Determine the stationary distribution T. (Hint: there are two approaches. One is to solve T P and ( 1 in general for doubly stochastic matrices. The other is to first solve a few examples, then make an educated guess...
2. A Markov chain is said to be doubly stochastic if both the rows and columns of the transition matrix sum to 1. Assume that the state space is {0, 1,....m}, and that the Markov chain is doubly stochastic and irreducible. Determine the stationary distribution T. (Hint: there are two approaches. One is to solve T P and ( 1 in general for doubly stochastic matrices. The other is to first solve a few examples, then make an educated guess...
Suppose Xn is a Markov chain on the state space S with transition probability p. Let Yn be an independent copy of the Markov chain with transition probability p, and define Zn := (Xn, Yn). a) Prove that Zn is a Markov chain on the state space S_hat := S × S with transition probability p_hat : S_hat × S_hat → [0, 1] given by p_hat((x1, y1), (x2, y2)) := p(x1, x2)p(y1, y2). b) Prove that if π is a...
Let α and β be positive constants. Consider a continuous-time Markov chain X(t) with state space S = {0, 1, 2} and jump rates q(i,i+1) = β for0≤i≤1 q(j,j−1) = α for1≤j≤2. Find the stationary probability distribution π = (π0, π1, π2) for this chain.
Suppose that we have a finite irreducible Markov chain Xn with stationary distribution π on a state space S. (a) Consider the sequence of neighboring pairs, (X0, X1), (X1, X2), (X2, X3), . . . . Show that this is also a Markov chain and find the transition probabilities. (The state space will be S ×S = {(i,j) : i,j ∈ S} and the jumps are now of the form (i, j) → (k, l).) (b) Find the stationary distribution...
. For a discrete time Markov chain with transition matrix P (py), explain what is meant by (1) global balance; (ii) local balance; (i) doubly stochastioc; (iv) birth-death Markov chain.
Let P be the n*n transition matrix of a Markov chain with a finite state space S = {1, 2, ..., n}. Show that 7 is the stationary distribution of the Markov chain, i.e., P = , 2hTi = 1 if and only if (I – P+117) = 17 where I is the n*n identity matrix and 17 = [11...1) is a 1 * n row vector with all components being 1.
1. Consider a time-homogeneous Markov chain X)n, such that P= 2 a) Calculate p12(2) b) Assuming Xo 1 (with probability 1), find the probability that Xn will reach state 2 before it reaches state 4 c) Find msz. d) Is the chain periodic? Irreducible? e) Find the stationary distribution f Approximate the probability that X0 1 g) Find the mean recurrence time for state 1