The appearance of an object of interest in a region of area $x$ units$^2$ follows a (two-dimensional) spatial Poisson process $N$ with constant rate $\lambda$ units$^{-2}$. An observer, who wishes to witness the object, can only view a fixed amount of total area, say $m$ units$^2$, in a single trip.
If distinct areas are visited in each trip, how many trips should the observer expect to take before they have witnessed the object $k$ times?
I am not sure how to formulate this problem. I can denote $B_i$ as the region visited in the $i$th trip. If I then denote the total area visited over all trips as $B$, I can write it like so: $$ B = \bigsqcup_{i=1}^{\ell}{B_i} $$
Where $\ell$ is the quantity that I want. Specifically, I believe I need the expected value of $\ell$, given that $N(B)=k$. Do you have any hints as to how I can properly formulate this question?
The probability of observing the object on the $i$th trip is equal to $$P(N(B_i) \ge 1) = 1-P(N(B_i) = 0) = 1 - e^{-\lambda \text{area}(B_i)}$$ since the areas are disjoint, and for a Poisson process the number of objects in disjoint intervals are independent, the number of trips till you observe the object once follows the distribution $$\text{Geometric}(1-e^{-\lambda m})$$ (we assumed that the $B_i$ have area $m$). Therefore the number of trips till you observe $k$ objects is the sum of independent geometric random variables, which is a distribution with a name that I'll let you research.