Statistics - MathematicalStatistics

Question 1: Represent this markov chain as I-P where P is the stochastic matrix, and I is the identity matrix.

$\begin{bmatrix}1/2&0&1/2&0&0\\0&1&0&0&0\\0&0&1&0&0\\0&0&0&1&0\\0&0&0&0&1\end{bmatrix}$ $\begin{bmatrix}1&-1/2&0&-1/2&0\\0&1&-1&0&0\\0&-1&1&0&0\\0&0&0&1&-1\\0&0&0&-1&1\end{bmatrix}$ $\begin{bmatrix}0&1/2&0&1/2&0\\0&0&1&0&0\\0&1&0&0&0\\0&0&0&0&1\\0&0&0&1&0\end{bmatrix}$ $\begin{bmatrix}1/2&0&-1/2&0&0\\0&1&0&0&0\\0&0&1&0&0\\0&0&0&1&0\\0&0&0&0&1\end{bmatrix}$

Question 2: Set up the system of equations where A is the previous matrix $A\mathbf{x} = \mathbf{b}$, where $\mathbf{x} = [x_1, x_2, x_3, x_4, x_5]^T$ represents the probability of reaching Node 4 starting from each node. Set $\mathbf{b} = [0, 0, 0, 0, 1]^T$ and solve for $\mathbf{x}$.

[1,0,0,1,1] [1/2,0,0,1,1] [1,1,1,0,0] [1/2,1,1,0,0]

Question 3: What does the previous questions' 0 node solution mean for the expected time to get from node 0 to node 4?

It takes an expected 2 steps to reach node 4 from node 0 It is impossible to reach node 4 from node 0 It takes an expected 1/2 steps to reach node 4 from node 0 The expectation will go to infinity as the probability is not 1

Question 4: What property do the two sets which cannot reach each other have when represented as a matrix?

The steps it will take to reach the other set is -1 The nodes probabilities are always 0 The probability of reaching the other set 1 The node sets sum are linearly independent of each other

Question 5: What does this property tell you about probabilities and expectations of the 2 sets of nodes?

The expectation of any node outside of these sets with positive probability for each will be finite The expectation from one to the other will always be finite The probability of getting from one set to the other is anywhere from 0 to anything below 1 The expectation of any node outside of these sets with positive probability for each will be undefined

Question 6: Which nodes have a finite expectation towards all other nodes in this new Markov Chain?

Node 0 Every Node Node 1, Node 0 None

Question 7: How does an extremely low probability differ from 0?

It allows for a defined expectation over time It has no effect on the Markov Chain It is effectively the same as 0 It guarantees reaching the state eventually in a finite amount of time

Question 8: Ignore previous markov chains, and consider exclusively the I matrix, can each node represented by this matrix reach the others?

Lower index Nodes can reach higher index nodes Each node loops to itself Each node can reach each other Higher index Nodes can reach lower index nodes

Question 9: What does it mean for expectations if the probability of a node to another is between 0 and 1

The expectation will be finite The expectation will be undefined All of the above CAN be true The expectation will be multiplied by a weight of p the probability

Question 10: What does it mean for expectations if the probability of a node to another is 1

The expectation will be finite The expectation is always 0 None of the above The expectation will be undefined

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