Question

Suppose Alice is sitting at a circular table with 4 chairs labeled {1, 2, 3, 4} and sitting initially at a random chair. Every minute she moves to her left or right at random with equal probability. Consider the Markov chain associated to the sequence of her positions X0, X1, . . . .

1. Write the state space, the distribution of X0 and the distribution of X1.

2. Write the transition matrix.

3. Assume she is at chair one at time t = 0. Write an expression (in terms of numbers Pn, do not evaluate) for the probability that she will be at 1 again at t = 4 and at a

different chair at t = 12.

Suppose Alice is sitting at a circular table with 4 chairs labeled 1, 2,3, 4) and sitting initially at a random chair. Every minute she moves to her left or right at random with equal probability. Consider the Markov chain associated to the sequence of her positions Xo, Xi, Write the state space, the distribution of Xo and the distribution of X. Write the transition matrix. . .. . Assume she is at chair one at time t = 0, Write an expression (in terms of numbers Pn, do not evaluate) for the probability that she will be at 1 again at t -4 and at a different chair at t 12.

Answer 1

1. As there are 4 chairs so 4 positions or 4 states, namely state 1, state 2, state 3, and state 4. Thus a state variable has 4 coordinates To start with at t-0 Alice is at chair 1 thus xo = 0 Also the chairs are in circular position as shon in the figure below. Thus from state 1 she can move to state 2 or state 3 with equal probabilty that is 0.5 for each state. Similarly if she is at state 2 then she can move to state 1 or state 3 with equal probabilty that is 0.5 for each state and so on. Thus the transition matrix P and the initial state are as shorbelow.

0.5 Transition matrix From To 1 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 X1 0.5 0 0.5 0 0 0.5 0 0.5 0.5 0 0.5 0 4 0.5 0 0.5 0 0 0.5 0.5 0.5 0.5 0.5 0.5

Assume she is at chair one at time t 0. Write an expression (in terms of numbers Pn, do not evaluate) for the probability that she will be at 1 agairn at t = 4 and at a different chair at t = 12. At t-4, the probability that she will be at 1 again os 0.5, and the probability that she will be at a different chair at t 12 is 0.5. X1 X3 X7 X10 X11 0.5 0 0.5 0 0.5 0.5 0 0.5 0.5 0.5 0.5 0 0.5 0 0.5 0.5 0 0.5 0 0.5 0.5 0.5 0 0.5 0.5 0.5 0.5 0.5 0.5 0 0.5 0.5 0.5 0 0.5 0.5 0.5 0.5 0.5 0.5 0.5 This we have obtained by repeatitive multiplications. Like PXi X(i+1)