Question

Consider a discrete time Markov chain with state space \(S=\{1,2\}\) and the transition probability diagram. If \(\pi=(\pi_1,\pi_2)\) is the stationary distribution of the Markov chain, then which one of the following statements is true?

• A. \((\pi_1,\pi_2)=\left(\frac{1}{2},\frac{1}{2}\right)\)
• B. \((\pi_1,\pi_2)=\left(\frac{2}{5},\frac{3}{5}\right)\)
• C. \((\pi_1,\pi_2)=\left(\frac{1}{4},\frac{3}{4}\right)\)
• D. \((\pi_1,\pi_2)=\left(\frac{1}{3},\frac{2}{3}\right)\)

Hint:

For a stationary distribution, always solve \(\pi P=\pi\) together with \(\pi_1+\pi_2+\cdots+\pi_n=1\).

Answer 1

The Correct Option is B

Step 1: Write the transition probability matrix.
From the diagram, the transition probabilities are
\[ P_{11}=0.25,\quad P_{12}=0.75 \] and
\[ P_{21}=0.5,\quad P_{22}=0.5 \]
Therefore, the transition matrix is
\[ P= \begin{pmatrix} 0.25 & 0.75\\ 0.5 & 0.5 \end{pmatrix} \]

Step 2: Use the stationary distribution condition.
The stationary distribution \(\pi=(\pi_1,\pi_2)\) satisfies
\[ \pi P=\pi \] and
\[ \pi_1+\pi_2=1 \]

Step 3: Form the first stationary equation.
Using the first component of \(\pi P=\pi\), we get
\[ \pi_1=0.25\pi_1+0.5\pi_2 \]
Rearranging,
\[ \pi_1-0.25\pi_1=0.5\pi_2 \] \[ 0.75\pi_1=0.5\pi_2 \]
Thus,
\[ \pi_2=\frac{0.75}{0.5}\pi_1 \] \[ \pi_2=1.5\pi_1 \]

Step 4: Use the total probability condition.
Since
\[ \pi_1+\pi_2=1 \] substitute
\[ \pi_2=1.5\pi_1 \]
Then,
\[ \pi_1+1.5\pi_1=1 \] \[ 2.5\pi_1=1 \] \[ \pi_1=\frac{1}{2.5}=\frac{2}{5} \]

Step 5: Find \(\pi_2\).
\[ \pi_2=1-\pi_1 \] \[ \pi_2=1-\frac{2}{5} \] \[ \pi_2=\frac{3}{5} \]

Step 6: Final conclusion.
Hence, the stationary distribution is
\[ \pi=\left(\frac{2}{5},\frac{3}{5}\right) \]
Therefore, the correct answer is
\[ \boxed{(B)} \]