Consider the following Exercise
Three out of $4$ trucks passing under a bridge are followed by a car, while $1$ out of every $5$ cars passing under that same bridge is followed by a
truck.a) Show that this is a transition matrix and write down the transition matrix.
b) Compute the stationary distribution.
c) What fraction of vehicles passing under the bridge are trucks ?
d) Given that the last vehicle seen was a truck, how long does it take on average until the next truck is seen under that same bridge ?
MY ANSWER
a) If we denote by $C$ and $T$ the states, we have $$\mathbb{P}=\begin{pmatrix} 4/5 & 1/5 \\ 3/4 & 1/4 \end{pmatrix}.$$
b) We know that the stationary distribution can be find by $\pi=\pi \mathbb{P}$. Thus, $\det(\mathbb{P}P^T – \mathbb{I})=0$ gives us the solution.
c) Thus, $4$ out of $19$ vehicles are trucks in the long-term.
QUESTION
d) I do not know why my professor here solves the following system $$\begin{cases}\mu_C = 1+ (4/5)\mu_C \\ \mu_T=1+(3/4)\mu_C\end{cases}$$
where $\mu_T$ and $\mu_C$ are the mean returning times from $T$ to $T$ and from $C$ to $C$ respectively. I do not know where this system of equations come from.
The only formula I know for the mean recurrent time of an state is the following $$\mu_j =\sum_{n=1}^{\infty} f_{j,j}^{(n)}$$
where $f_{i,j}^{(n)}$ is defined as $P(X_1\neq j,…,X_{n-1}\neq j, X_n =j\, |\, X_0=i)$.
Best Answer
If we denote by $\mu_i = \mathbb{E}[T\,|\, X_0=i]$ the returning time to state $i$, then we have that $$\mu_i = 1+\sum_{j\in S} \mu_j p_{i,j}$$ where in the sum we set $\mu_j=0$ if and only if the last state seen is $j$. Thus, in my case, we have
$$\mu_C=1+(4/5)\mu_C + (1/5) \cdot 0$$ $$\mu_T=1+(3/4)\mu_C + (1/4) \cdot 0$$
hence $$\mu_C=1+(4/5)\mu_C$$ $$\mu_T=1+(3/4)\mu_C .$$