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[saifh]

Hi there,

I'm currently doing one of the questions off a practice sheet given to us at uni, however am stuck due to not fully understanding the concept of the question. I've tried attempting it different ways but just cannot find a place to start. Can anyone please help me out here? I've uploaded the question as an image:

View attachment 4823
Thanks!

 

[Opalg]

Hi saifh, and welcome to MHB!

Think of the matrix $\mathbf{P}$ like this: $$\begin{matrix}&&\text{from} \\ &&\begin{matrix}1\;\; &2&\;\; 3\end{matrix} \\ \text{to} & \begin{matrix}1 \\ 2\\ 3\end{matrix} & \begin{bmatrix}0.5 & 0.4 & 0.6 \\ 0.2&0.2 & 0.3 \\ 0.3 & 0.4 & 0.1\end{bmatrix} \end{matrix}.$$ In other words, the rows and columns are each numbered by the three states $1$, $2$, $3$, and the element in row $i$ and column $j$ of the matrix represents the probability of transition from state $j$ to state $i$. So for example $\mathbf{P}_{2,1}$, the element in row $2$ and column $1$ (which is equal to $0.2$) tells you the probability of transition from state $1$ to state $2$. That should be enough help for you to do part (i).

For part (ii), draw a diagram with three regions labelled $1$, $2$, $3$ and link them with arrows going from each region to each region (including circular arrows going from a region to itself). Then label each arrow with the probability of transition from the state at the tail of the arrow to the to state at the head of the arrow.

For (iii), look in your notes or textbook to see the definitions of regular and stochastic, and check that the matrix $\mathbf{P}$ satisfies them.

 

[saifh]

Hi saifh, and welcome to MHB!

Think of the matrix $\mathbf{P}$ like this: $$\begin{matrix}&&\text{from} \\ &&\begin{matrix}1\;\; &2&\;\; 3\end{matrix} \\ \text{to} & \begin{matrix}1 \\ 2\\ 3\end{matrix} & \begin{bmatrix}0.5 & 0.4 & 0.6 \\ 0.2&0.2 & 0.3 \\ 0.3 & 0.4 & 0.1\end{bmatrix} \end{matrix}.$$ In other words, the rows and columns are each numbered by the three states $1$, $2$, $3$, and the element in row $i$ and column $j$ of the matrix represents the probability of transition from state $j$ to state $i$. So for example $\mathbf{P}_{2,1}$, the element in row $2$ and column $1$ (which is equal to $0.2$) tells you the probability of transition from state $1$ to state $2$. That should be enough help for you to do part (i).

For part (ii), draw a diagram with three regions labelled $1$, $2$, $3$ and link them with arrows going from each region to each region (including circular arrows going from a region to itself). Then label each arrow with the probability of transition from the state at the tail of the arrow to the to state at the head of the arrow.

For (iii), look in your notes or textbook to see the definitions of regular and stochastic, and check that the matrix $\mathbf{P}$ satisfies them.

Cheers for the help, I'll reply back here or PM if I still can't do this.