Statistics - MathematicalStatistics

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

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

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

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 node sets sum are linearly independent of each other The probability of reaching the other set 1

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 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 The expectation from one to the other will always be finite

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

Every Node None Node 1, Node 0 Node 0

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

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

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

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

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

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

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

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

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