Statistics - MathematicalStatistics

Question 1: What is the stochastic matrix of the Markov Chain pictured here:

No stochastic matrix exists $\begin{bmatrix}1/2&1/2&0\\1/2&1/2&0\\0&0&1\end{bmatrix}$ $\begin{bmatrix}1/3&1/3&1/3\\1/3&1/3&1/3\\1/3&1/3&1/3\end{bmatrix}$ $\begin{bmatrix}0&1/2&1/2\\0&1/2&1/2\\0&0&1\end{bmatrix}$

Question 2: What state will you be in after 100 steps in this Markov Chain?

2 No way to tell 1 0

Question 3: What is the probability of being in state 1 after 3 steps?

1/8 1 1/4 1/2

Question 4: What is the probability of being in state 1 after n steps?

$(1/2)^n$ 0 1 $(1/2)^{(n-1)}$

Question 5: In expectation how many steps will it take for nodes 0, and 1 to reach node 2?

For node 0:1.5, For node 1:2 For node 0:2, For node 1:1.5 For node 0:1, For node 1:1 For node 0:2, For node 1:2

Question 6: What is the stochastic matrix of the Markov Chain pictured here:

$\begin{bmatrix}0&1/2&0&1/2&0\\0&0&1&0&0\\0&1&0&0&0\\0&0&0&0&1\\0&0&0&1&0\end{bmatrix}$ $\begin{bmatrix}1/5&1/5&1/5&1/5&1/5\\1/5&1/5&1/5&1/5&1/5\\1/5&1/5&1/5&1/5&1/5\\1/5&1/5&1/5&1/5&1/5\\1/5&1/5&1/5&1/5&1/5\end{bmatrix}$ $\begin{bmatrix}1/2&0&1/2&0&0\\0&1&0&0&0\\0&0&1&0&0\\0&0&0&1&0\\0&0&0&0&1\end{bmatrix}$ No stochastic matrix exists

Question 7: Is it possible to reach node 2 from node 4?

Yes, with a large probability Guranteed No With a small probability

Question 8: What is the probability of reaching node 2 in infinite steps from node 0?

1/2 1/4 1 0

Question 9: What is the expected number of steps it will take to reach node 2 from node 1?

1 1/2 2 undefined

Question 10: What is the expected number of steps it will take to reach node 2 from node 0?

undefined 1/2 1 2

Quiz Questions