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(b) Let {Xn : n E N} b...
Let X0,X1,... be a Markov chain whose state space is Z (the integers). Recall the Markov property: P(Xn = in | X0 = i0,X1 = i1,...,Xn−1 = in−1) = P(Xn = in | Xn−1 = in−1), ∀n, ∀it. Does the following...
Let X0,X1,... be a Markov chain whose state space is Z (the
integers).
Recall the Markov property: P(Xn = in | X0 = i0,X1 = i1,...,Xn−1
= in−1) = P(Xn = in | Xn−1 = in−1), ∀n, ∀it. Does the following
always hold: P(Xn ≥0|X0 ≥0,X1 ≥0,...,Xn−1 ≥0)=P(Xn ≥0|Xn−1 ≥0)
?
(Prove if “yes”, provide a counterexample if “no”)
Let Xo,Xi, be a Markov chain whose state space is Z (the integers). Recall the Markov property: P(X,-'n l Xo-io, Xi...
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 transi...
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...
Consider a process {Xn, n = 0,1, ... }, which takes on the values 0,1, or...
Consider a process {Xn, n = 0,1, ... }, which takes on the values 0,1, or 2. Suppose P{Xn+1 = ||Xn = i, Xn-1 = in-1,..., X0 = i +0} = when n is even when n is odd where - P = - Phl = 1, i = 0,1,2. Is {Xn, n = 0,1,... } a time-homogeneous Markov chain? If not, then show how, by enlarging the state space, we may transform it into a time- homogeneous Markov chain.
4. Let Z1, Z2,... be a sequence of independent standard normal random variables. De- fine Xo 0 and n=0, 1 , 2, . . . ....
4. Let Z1, Z2,... be a sequence of independent standard normal random variables. De- fine Xo 0 and n=0, 1 , 2, . . . . TL: n+1 , The stochastic process Xn,n 0, 1,2,3 is a Markov chain, but with a continuous state space. (a) Find EXn and Var(X). (b) Give probability distribution of Xn (c) Find limn oo P(X, > є) for any e> 0. (d) Simulate two realisations of the Markov process from n = 0 until...
Help please! Let {Zn}n=0 be iid. Bernoulli random, vari- + Zn. ables with PZ-0] = p...
Help please!
Let {Zn}n=0 be iid. Bernoulli random, vari- + Zn. ables with PZ-0] = p and PZ-1] 1-p. Define Sn-Zo + Which of the following processes is a Markov chain? 1. An S For each process that is a Markov chain, find its transition matriz. For each process Xn E An, Bn, Cn, Dn that is not a Markov chain, find a pair of states i and j so that P[Xn+ 1 = ilXn-j, Xn-1-k] depends on k.
1. Let {Xn, n 2 0 be a Markov Chain with state space S. Show that...
1. Let {Xn, n 2 0 be a Markov Chain with state space S. Show that for any n, m-1 and JAn+m, . . . , İn+1,in-1 , . . . , io є S.
Suppose that we have a finite irreducible Markov chain Xn with stationary distribution π on a...
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...
Let Z1, Z2, . . . be a sequence of independent standard normal random variables. Define X0 = 0 and Xn+1 = (nXn + (Zn+1))...
Let Z1, Z2, . . . be a sequence of independent standard normal random variables. Define X0 = 0 and Xn+1 = (nXn + (Zn+1))/ (n + 1) , n = 0, 1, 2, . . . . The stochastic process {Xn, n = 0, 1, 2, } is a Markov chain, but with a continuous state space. (a) Find E(Xn) and Var(Xn). (b) Give probability distribution of Xn. (c) Find limn→∞ P(Xn > epsilon) for any epsilon > 0.
Consider the process E here Xn is the outcome of a die on the nth roll at XnEN is a Markov chain. (b) Determine the state space S and the transition matrix P (with, as usual, reasoning Consider...
Consider the process E here Xn is the outcome of a die on the nth roll at XnEN is a Markov chain. (b) Determine the state space S and the transition matrix P (with, as usual, reasoning
Consider the process E here Xn is the outcome of a die on the nth roll at XnEN is a Markov chain. (b) Determine the state space S and the transition matrix P (with, as usual, reasoning
This is for Stochastic Processes Let Xo, Xi,... be a Markov chain whose state space is Z (the integers). Recall the Markov property: P(X, _ in l Xo-to, X1-21, , Xn l-an l)-P(Xn-in l x, i-İn 1), Vn, V...
This is for Stochastic Processes
Let Xo, Xi,... be a Markov chain whose state space is Z (the integers). Recall the Markov property: P(X, _ in l Xo-to, X1-21, , Xn l-an l)-P(Xn-in l x, i-İn 1), Vn, Vil. Does the following always hold: (lProve if "yes", provide a counterexample if "no")
Let Xo, Xi,... be a Markov chain whose state space is Z (the integers). Recall the Markov property: P(X, _ in l Xo-to, X1-21, , Xn l-an l)-P(Xn-in...
Let X0,X1,... be a Markov chain whose state space is Z (the
integers).
Recall the Markov property: P(Xn = in | X0 = i0,X1 = i1,...,Xn−1
= in−1) = P(Xn = in | Xn−1 = in−1), ∀n, ∀it. Does the following
always hold: P(Xn ≥0|X0 ≥0,X1 ≥0,...,Xn−1 ≥0)=P(Xn ≥0|Xn−1 ≥0)
?
(Prove if “yes”, provide a counterexample if “no”)
Let Xo,Xi, be a Markov chain whose state space is Z (the integers). Recall the Markov property: P(X,-'n l Xo-io, Xi...
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...
Consider a process {Xn, n = 0,1, ... }, which takes on the values 0,1, or 2. Suppose P{Xn+1 = ||Xn = i, Xn-1 = in-1,..., X0 = i +0} = when n is even when n is odd where - P = - Phl = 1, i = 0,1,2. Is {Xn, n = 0,1,... } a time-homogeneous Markov chain? If not, then show how, by enlarging the state space, we may transform it into a time- homogeneous Markov chain.
4. Let Z1, Z2,... be a sequence of independent standard normal random variables. De- fine Xo 0 and n=0, 1 , 2, . . . . TL: n+1 , The stochastic process Xn,n 0, 1,2,3 is a Markov chain, but with a continuous state space. (a) Find EXn and Var(X). (b) Give probability distribution of Xn (c) Find limn oo P(X, > є) for any e> 0. (d) Simulate two realisations of the Markov process from n = 0 until...
Help please!
Let {Zn}n=0 be iid. Bernoulli random, vari- + Zn. ables with PZ-0] = p and PZ-1] 1-p. Define Sn-Zo + Which of the following processes is a Markov chain? 1. An S For each process that is a Markov chain, find its transition matriz. For each process Xn E An, Bn, Cn, Dn that is not a Markov chain, find a pair of states i and j so that P[Xn+ 1 = ilXn-j, Xn-1-k] depends on k.
1. Let {Xn, n 2 0 be a Markov Chain with state space S. Show that for any n, m-1 and JAn+m, . . . , İn+1,in-1 , . . . , io є S.
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...
Consider the process E here Xn is the outcome of a die on the nth roll at XnEN is a Markov chain. (b) Determine the state space S and the transition matrix P (with, as usual, reasoning
Consider the process E here Xn is the outcome of a die on the nth roll at XnEN is a Markov chain. (b) Determine the state space S and the transition matrix P (with, as usual, reasoning
This is for Stochastic Processes
Let Xo, Xi,... be a Markov chain whose state space is Z (the integers). Recall the Markov property: P(X, _ in l Xo-to, X1-21, , Xn l-an l)-P(Xn-in l x, i-İn 1), Vn, Vil. Does the following always hold: (lProve if "yes", provide a counterexample if "no")
Let Xo, Xi,... be a Markov chain whose state space is Z (the integers). Recall the Markov property: P(X, _ in l Xo-to, X1-21, , Xn l-an l)-P(Xn-in...
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