aiX1, X ndependeni identically distrilnd random varialbkes taking val ues on 0, 1,2,.. with probabilities pi-P(X5....
(a) Suppose that Xi, X2,... are independent and identically distributed random variables each taking the value 1 with probability p and the value-1 with probability 1-p For n 1,2,..., define Yn -X1 + X2+ ...+Xn. Is {Yn) a Markov chain? If so, write down its state space and transition probability matrix. (b) Let Xı, X2, ues on [0,1,2,...) with probabilities pi-P(X5 Yn - min(X1, X2,.. .,Xn). Is {Yn) a Markov chain and transition probability matrix. be independent and identically distributed...
(a) Suppose that X1, X2,... are independent and identically distributed random variables each taking the value 1 with probability p and the value -1 with probability 1-p. For n = Yn-X1 + X2 + . . . + Xn. Is {Y, a Markov chain? If so, write down its state space and transition probability matrix 1, 2, . . ., denne
6. Suppose Xn is a two-state Markov chain with transition probabilities (Xn, Xn+1), n = 0, 1, 2, Write down the state space of the Markov chain Zo, Zi, . . . and determine the transition probability matrix.
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...
7. Define a Markov Chain on S-0,1,2,3,... with transition probabilities Pi,i+1 with 0<p < 1/2. Prove that the Markov Chain is reversible.
Consider a Markov chain with transition probabilities p(x, y), with state space S = {1, 2, . . . , 10}, and assume X0 = 3. Express the conditional probability P3(X6 =7, X5 =3 | X4 =1, X9 =3) entirely in terms of (if necessary, multi-step) transition probabilities.
Consider the Markov chain with state space {0, 1,2} and transition matrix(a) Suppose Xo-0. Find the probability that X2 = 2. (b) Find the stationary distribution of the Markov chain
Problem 3.3 (10 points) Consider a two-state continuous time Markov chain with state space 11,2) and transition function (a) Find P(X-21 Xo = 1]. (b) Find P[X5 1, X2 2 X1-1] Problem 3.3 (10 points) Consider a two-state continuous time Markov chain with state space 11,2) and transition function (a) Find P(X-21 Xo = 1]. (b) Find P[X5 1, X2 2 X1-1]
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.
3. Let U1, U2,. be a sequence of independent Ber(p) random variables. Define Xo 0 and Xn+1-Xn +2Un-1, 1,2,.. (a) Show that X, n 0,1,2, is a Markov chain, and give its transition graph. (b) Find EX and Var(X) c)Give P(X