# Expectation problem in Absorbing Markov Chain(exercise on Grinstead and Snell 11.2 18 )

Hi I encountered this problem. It took me quite long but I could not solve it. The problem is as follows: Assume that a student going to a recently established school in a university has, each year, a probability 0.1 of ﬂunking out, a probability 0.2 of having to repeat the year, and a probability 0.7 of moving on to the next year (in the fourth year, moving on means graduating).

1. Find how long a fresh student can expect to be in this school.

My idea is using conditional expectation(graduate or flunk out) but I can't solve the part for flunking out. It is in the book of Grinstead and Snell. Can any one help me?

If using the transition matrix, the problem will end up in trying to find $\sum_{i=1}^\infty P_1^i*i$ or something like that.(some entries in the ith power of the transition matrix.)

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You can begin by creating a transition matrix with 5 states (Freshman, Sophomore, Junior, seNior, and Gone), which would look like the following: $$\begin{bmatrix} &F &S &J &N &G \\ F &.2 &.7 &0 &0 &.1 \\ S& 0& .2& .7& 0& .1\\ J& 0& 0& .2& .7 & .1\\ N& 0& 0 & 0 & .2 & .8\\ G& 0& 0& 0& 0& 1 \end{bmatrix}$$
Notice that this has 4 transient states and 1 absorbing state (Gone), and is already in canonical form, with $$Q = \begin{bmatrix} .2 &.7 &0 &0 \\ 0& .2 & .7 &0 \\ 0& 0 & .2 &.7 \\ 0& 0 & 0 & .2 \end{bmatrix}$$
You can then calculate the fundamental matrix $N = (I-Q)^{-1}$. The sum of the each row of N tells you the expected number of years prior to absorption (beginning in that state). A convenient way to see this is to mutliply $N$ by the column vector $[1 1 1 1]^{T}$. Since you're interested in an incoming Freshman, you'll want the sum of the first row of $N$ (or the first entry in the column vector after multiplying by $[1 1 1 1]^{T}$). I get about 4.138 years, which should make sense, since the probability of repeating a grade (i.e., spending more than four years) is greater than the probability of flunking out early.