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: Read the matrix from the picture.
The two state chain has $P_{11}=0.25,\,P_{12}=0.75$ from state $1$ and $P_{21}=0.5,\,P_{22}=0.5$ from state $2$.

Step 2: Use balance between the two states.
In steady state the flow $1\to2$ equals the flow $2\to1$: $\pi_1(0.75)=\pi_2(0.5)$.

Step 3: Get the ratio.
This gives $\pi_2=1.5\,\pi_1$.

Step 4: Normalise.
Since $\pi_1+\pi_2=1$, we have $2.5\,\pi_1=1$, so $\pi_1=\dfrac25$ and $\pi_2=\dfrac35$.

Step 5: Conclude.
The stationary distribution is $\left(\dfrac25,\dfrac35\right)$, option (B).
\[ \boxed{\left(\frac25,\frac35\right)} \]