Imagine that there are $k$ distinguishable balls and $n$ distinguishable boxes. Each ball is placed into a random box. Next we calculate the maximum number of balls in a single box, which will be a value between $\left \lceil \dfrac{k}{n}\right \rceil$, when the balls are distributed almost evenly, and $k$, all balls in one box. Let's call this $X_{k,n}$
- What distribution does $X_{k,n}$ follow?
- I've encountered a situation where $k$ is about $100,000$ and $n$ is about $100$. How could I estimate the CDF for these particular values? What is $\mathbb{P}(X_{k,n}<2000)$?