A particle moves in a circle through points that have been marked 0, 1, 2, 3, 4 in a clockwise direction.order. The particle starts at point 0. At each step it has probability 0.35 of moving one point clockwise (0follows 4) and 0.65 of moving one point counter clockwise.Let Xn denote its location on the circle after step n.a). Construct the one step transition matrix for Xn.b) . Determine n-step transition matrix for n =5.
Question
A particle moves in a circle through points that have been marked 0, 1, 2, 3, 4 in a clockwise direction.order. The particle starts at point 0. At each step it has probability 0.35 of moving one point clockwise (0follows 4) and 0.65 of moving one point counter clockwise.Let Xn denote its location on the circle after step n.a). Construct the one step transition matrix for Xn.b) . Determine n-step transition matrix for n =5.
Solution
a) The one-step transition matrix P for Xn can be constructed as follows:
P = [0.65, 0.35, 0, 0, 0; 0.35, 0.65, 0.35, 0, 0; 0, 0.35, 0.65, 0.35, 0; 0, 0, 0.35, 0.65, 0.35; 0.35, 0, 0, 0.35, 0.65]
Each row of the matrix represents the current state (from 0 to 4), and each column represents the next state. The entries in the matrix represent the probability of transitioning from the current state to the next state.
b) The n-step transition matrix for n = 5 can be determined by raising the one-step transition matrix to the power of 5. This can be done using matrix multiplication. However, this calculation can be complex and is typically done using a computer or calculator with matrix operations.
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