MMiDS 7.3: Self-Assessment Quiz

Which of the following is the correct condition for a probability distribution $\pi =({\pi }_i{)}_{i=1}^n$ to be a stationary distribution of a Markov chain with transition matrix $P=(p_{i,j}{)}_{i,j=1}^n$?

• a) $\sum _{j=1}^n{\pi }_i{p}_{i,j}={\pi }_j$ for all $i\in [n]$
• b) $\sum _{i=1}^n{\pi }_i{p}_{i,j}={\pi }_j$ for all $j\in [n]$
• c) $\sum _{j=1}^n{\pi }_i{p}_{i,j}={\pi }_i$ for all $i\in [n]$
• d) $\sum _{i=1}^n{\pi }_i{p}_{i,j}={\pi }_i$ for all $j\in [n]$

Which of the following is the matrix form of the condition for a probability distribution $\pi$ to be a stationary distribution of a Markov chain with transition matrix $P$?

• a) $\pi P=\pi$
• b) $P\pi =\pi$
• c) $\pi P^T={\pi }^T$
• d) $P^T{\pi }^T={\pi }^T$

Consider a Markov chain with the following transition matrix:

$$P=\left(\begin{array}{cc}1/2& 1/2\\ 1/3& 2/3\end{array}\right)$$

Which of the following is a stationary distribution for this Markov chain?

• a) $(1/2,1/2)$
• b) $(1/3,2/3)$
• c) $(2/5,3/5)$
• d) $(1,0)$

A Markov chain is irreducible if:

• a) Every state communicates with every other state.
• b) There exists a state that communicates with every other state.
• c) The transition graph of the chain is strongly connected.
• d) Both a and c.

Consider the following transition graph of a Markov chain:

G = nx.DiGraph()
G.add_edges_from([(0, 1), (1, 2), (2, 0), (1, 1)])

Is this Markov chain irreducible?

• a) Yes
• b) No
• c) Not enough information
• d) The concept of irreducibility does not apply to this chain.

Consider the following transition graph of a Markov chain:

G = nx.DiGraph()
G.add_edges_from([(1, 2), (2, 1), (2, 3), (3, 3)])

Is this Markov chain irreducible?

• a) Yes
• b) No

According to the Existence of Stationary Distribution Theorem, an irreducible Markov chain on a finite state space:

• a) Has a unique stationary distribution with all entries being strictly positive.
• b) May have multiple stationary distributions, but all of them have strictly positive entries.
• c) Has a unique stationary distribution, but some entries may be zero.
• d) May not have any stationary distribution.

In an irreducible Markov chain, the left and right eigenvectors corresponding to eigenvalue 1 are:

• a) The same up to scaling.
• b) Always different.
• c) Transposes of each other.
• d) Not necessarily related to each other.

For an irreducible Markov chain, which of the following statements is true?

• a) All eigenvalues of the transition matrix have absolute value strictly less than 1.
• b) All eigenvalues of the transition matrix have absolute value less than or equal to 1, and eigenvalue 1 has geometric multiplicity 1.
• c) All eigenvalues of the transition matrix have absolute value less than or equal to 1, and eigenvalue -1 has geometric multiplicity 1.
• d) The transition matrix may have eigenvalues with absolute value greater than 1.

Given the transition matrix $P$ of a Markov chain, which method can be used to numerically find the stationary distribution?

• a) Solving the linear system $\pi (P-I)=0$.
• b) Computing the eigenvector corresponding to the eigenvalue 1 of $P$.
• c) Both a and b.
• d) None of the above.