Each morning a runner leaves his house and goes for a jog. He is equally likely to leave either from his front or back door. Upon leaving the house, he chooses a pair of sports shoes (or goes for a jog barefoot if there are no shoes at the door from which he left). On his return he is equally likely to enter, and leave his sports shoes, either by the front or back door.
I want to arrive at the transition matrix of the process $X_n$, where $X_n$ represents the number of shoes at the front door.
I am missing some logic and I could not arrive at the following transition matrix.
\begin{bmatrix}
0.75 & 0.25 & 0 & \dots & 0 \\
0.25 & 0.5 & 0.25 & \dots & 0 \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
0 & 0 & 0 & \dots & 0.75
\end{bmatrix}
Can somebody help me understand this clearly? If possible, with the help of a probability tree?
Best Answer
n = total number of shoes
The table shows the new number of shoes at the front door when returning runners:
I would say, transition probabilities:
$p_{0,0}= 3\cdot 0.25 = 0.75, \quad p_{0,1}=0.25$
$p_{i,i-1}=0.25,\quad p_{i,i}=2\cdot 0.25=0.50,\quad p_{i,i+1}=0.25, \quad i\in\langle 1, n-1 \rangle$
$p_{n,n-1}=0.25, \quad p_{n,n}= 3\cdot 0.25 = 0.75$
$\Rightarrow\,\,$ Your transition matrix.