Question

The PSTAT Department at UCSB has developed a Markov model to simulate graduation rates in their programs. Students might drop out, repeat a year, or move on to the next year until they graduate. Students have a 3% chance of repeating their current year. First-years and sophomores have a 6% chance of dropping out. For juniors and seniors, the drop-out rate is 4%. Provide the state space S and the transition probability matrix for this Markov chain. Also draw the associated transition graph.

Answer 1

Answer:

State space

$S=\{F,So,J,Sr,G,D\}$

Transition probability matrix

$\begin{bmatrix} & F & So&J&Sr&G&D\,\\ F &0.03&0.91&0&0&0&0.06\,\\So&0&0.03&0.91&0&0&0.06\\J&0&0&0.03&0.93&0&0.04\\ Sr&0&0&0&0.03&0.93&0.04\\G&0&0&0&0&1&0\\D&0&0&0&0&0&1\end{bmatrix}$

The transition graph is attached.

Step-by-step explanation:

The state space for this Markov chain is all the possible states that the variable can take.

In this case the student can be: Freshman, Sophomore, Junior, Senior, Graduate and Drop out

$S=\{F,So,J,Sr,G,D\}$

The transition probability matrix for this state space is:

$\begin{bmatrix} & F & So&J&Sr&G&D\,\\ F &0.03&0.91&0&0&0&0.06\,\\So&0&0.03&0.91&0&0&0.06\\J&0&0&0.03&0.93&0&0.04\\ Sr&0&0&0&0.03&0.93&0.04\\G&0&0&0&0&1&0\\D&0&0&0&0&0&1\end{bmatrix}$

The transition graph is attached.