Discussion
Suppose \(\{X_{n}, n \ge 0\}\) is a Markov chain with state space S = {1, 2, 3} and the transition probability matrix P given by (rows and columns indexed by the states 1, 2, 3) :
\[ P = \begin{bmatrix} \frac{2}{5} & \frac{3}{5} & 0 \\ \frac{3}{8} & \frac{5}{8} & 0 \\ \frac{1}{6} & \frac{1}{3} & \frac{1}{2} \end{bmatrix} \]
Suppose X0 = 3. Then realized values of X1 and X2 corresponding to random numbers 0.36 and 0.64 respectively, from U(0, 1) distribution are :
Answer: A
The realized values are X1 = 2 and X2 = 2.
To simulate a step, compare the uniform number with the cumulative probabilities of the current row.
From \(X_{0} = 3\), row 3 is \((\frac{1}{6}, \frac{1}{3}, \frac{1}{2})\), with cumulative values 0.1667, 0.5, 1. Since \(0.1667 \lt 0.36 \le 0.5\), \(X_{1} = 2\).
From state 2, row 2 is \((\frac{3}{8}, \frac{5}{8}, 0)\), with cumulative values 0.375, 1, 1. Since \(0.375 \lt 0.64 \le 1\), \(X_{2} = 2\).
Reading 0.36 against row 1 or 0.64 against row 3 gives the other combinations, which use the wrong rows.
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