Question

Let \( \{X_n\}_{n \geq 0} \) be a time-homogeneous discrete time Markov chain with state space \( \{0, 1\} \) and transition probability matrix

\[ P = \begin{bmatrix} 0.25 & 0.75 \\ 0.75 & 0.25 \end{bmatrix}. \]

If \( P(X_0 = 0) = P(X_0 = 1) = 0.5 \), then

\[ \sum_{k=1}^{100} E[(X_{2k})^2] \]

equals _________ (round off to 2 decimal places).

Hint:

For a Markov chain, the long-term expected value of any state can be calculated using the stationary distribution.

Answer 1

Correct Answer: 50

Since \( X_n \) is a Markov chain with transition probabilities, the expected value \( E[(X_{2k})^2] \) represents the probability of \( X_{2k} = 1 \). The transition probabilities lead to a steady-state distribution of \( \frac{1}{2} \) for both states \( 0 \) and \( 1 \). Therefore, \[ \sum_{k=1}^{100} E[(X_{2k})^2] = 100 \times \frac{1}{2} = 50. \] Thus, the sum equals 50.