Consider the numbers 1 2 25 written around on a circle one

Consider the numbers {1, 2, .... 25} written around on a circle, one after the another (so that, in particular, 1 is adjacent to 2 and 25). Consider a Markov chain (X_n)_n greaterthanorequalto 0 that, at each time, jumps with equal probability to one of the two adjacent numbers. a) What is the expected number of steps that X_n will take to return to its starting position? b) What is the probability that X_n will visit all the other states before returning to its starting position?

Solution

ANSWER:-

One way of getting the result is to say that ,after the move to state 2 and the cut each further step changes by +1 with probability 1/2 or by -1 with probability 1/2.

so the expected value of the state remains 2 from this point forward.

Since you have absorbing states one of these events must happen

P₂(V₂₅ > V₁ ) + P₂(V₂₅ < V₁ ) = 1

and to keep the expectation a constant 2 you have

1* P₂(V₂₅ > V₁ ) + 25* P₂(V₂₅ < V₁ ) = 2

and so by substitution 1* (1- P₂(V₂₅ < V₁ )) + 25* P₂(V₂₅ < V₁ ) = 2

(25-1) * P₂(V₂₅ < V₁ ) = 2-1

P₂(V₂₅ < V₁ ) = 1/24.