Consider a train that starts from the LA-Union station. Let $I_{m}$ denote the number of people that get into the train from the station $m$. Assume that $I_{m}$ are independent and Poisson with parameter $\lambda_{m} .$ Each person entering from the station $m$ will, independent of everything else, get off at the station $n$ with probability $P_{m, n}$. Also, $\sum_{n>m} P_{m, n}=1$. Let $G_{n}$ be the number of people getting off the train at the station $n$.
- Calculate $\mathbb{E}[G_n]$.
- What is the distribution of $G_n$?
- What is the joint distribution of $G_n$ and $G_k$?
My solution is as follows.
Let $ N_{i,j} $ be the number of people that get on at station $ i $ and get off at station $ j $($ i < j $), then $ N_{i,j} = P_{i,j}I_i $. Since $ G_n $ is the number of people getting off teh train at the station $ n $, we have $ G_n = \sum_{m<n} P_{m,n}I_m $, thus \begin{equation} \mathbb{E}(G_n) = \mathbb{E}\left(\sum_{m=1}^{n-1} N_{m,n} \right) = \sum_{m=1}^{n-1} \mathbb{E}(N_{m,n} ) = \sum_{m=1}^{n-1} \mathbb{E}(P_{m,n}I_m ) = \sum_{m=1}^{n-1} P_{m,n} \lambda_m. \end{equation} $ G_n =\sum_{m<n} P_{m,n}I_m $ is the weighted sum of independent Poisson random variables, But I don't know how to give the distribution of $G_n$, such as the probability density function of $G_n$ and the joint distribution of $G_n$ and $G_k$.