So I am currently in the process of learning basic distributions of random variables and I have been trying to understand the following scenario (distilled for sake of brevity):
The number of people who enter an elevator on the ground floor is a Poisson random variable with mean $10$. If there are $N$ floors above the ground floor and if each person is equally likely to get off at any one of these $N$ floors, independently of where the others get off, compute the expected number of stops that the elevator will make before discharging all of its passengers.
So, my thought process is as follows.
Let $X=$ number of people who enter the elevator. $X \sim Poisson(10)$.
Let $Y=$ the number of stops it takes in total.
I define the indicator variable $I_n$ such that it equals $1$ if the elevator stops at a given floor and $0$ otherwise.
Then, $Y = I_1 + … + I_N$.
From what I understand, the goal is to find $E[Y]$ since this essentially gives the expected number of times the elevator will stop.
Now, $E[Y] = \sum_{n=1}^N E[I_n] = NE[I_n]$.
Thus, I need to find $I_n$ but this is where I am stuck.
I have attempted to condition on $X$ (the number of people who entered the elevator) since naturally, the number of stops it takes depends on the number of people that entered to begin with, but then I am not sure how to go after defining $I_n$ in terms of the conditional expectation: $E[I_n] = E[E[I_n | X = m]]$. I know that I need to find $E[I_n | X = m]$ somehow, but I am not sure how to do so.
This is not a homework question and I am really trying to understand the properties of expectation and how to work with distributions on a deeper level rather than just memorizing a strategy for solving these problems and I would greatly appreciate any help.
Best Answer
The number of people getting off on a given floor is a Poisson variable with mean $\frac{10}N$. Thus the probability of not stopping at a given floor is $\mathrm e^{-10/N}$, and the expected number of stops is $N\left(1-\mathrm e^{-10/N}\right)=10-\frac{50}N+O\left(N^{-2}\right)$.