Markov Chain Question

Hi Guys,

Can anyone answer part (iii) from the attached picture. I can't work it out.

Thanks

N

 

The transition matrix is
P = \begin{matrix} 1/3 & 2/3 & 0 \\ 1/3 & 0 & 2/3 \\ 0 & 1/3 & 2/3 \end{matrix}

Let m_i be the expected number of policy years until the new policyholder reaches level 2 given currently he is in level i.
So we can form the following equations:
$$m_0 = 1 + 1/3 m_0 + 2/3 m_1$$

$$m_1 = 1 + 1/3 m_0 + 2/3 m_2$$

Note that we are adding 1 as the policyholder changes levels only after 1 year. Also m_2 = 0.
Solving the equations we get m_0 = 3.75 years. This should be the answer. Let me know if I have erred somewhere.

 

Probability of no accident is 1/3. So, I guess the values 1/3 and 2/3 in the matrix should be swapped.

 

My bad, I made a mistake there.