I have created the following example to ensure my Poisson Process methodology is correct for a counting process that refreshes at every time interval.
Example Chewing gum is dropped on a specific area of pavement at a rate of $10$ per year as a Poisson Process. At the exact same time each year, a cleaner arrives to clean the street. He will remove every single bit of gum form this specific area of pavement. We assume the time it takes him to do this is ~0 years and thus, negligible. How many pieces of gum can he expect to remove and what is the variance?
Let $\{N_t: t \ge 0\}$ be a Poisson Process with rate $\lambda$ where in our example $\lambda=10$ and $N_t$ represents the gum found by a clearer after $t$ years.
Question Since the cleaner arrives faithfully every year, am I correct in setting $N \equiv N_1 = \{N_t: t \ge 0\}$?
Furthermore, is following then trivially true for every year? $$ V(N) = E(N) = \lambda\\ $$