I frequently run into this question while modeling processes. I am wondering if there is a general solution or approximation.
The question I run into is: For a given distribution with a finite number of points, how much of the distribution have I explored after viewing $n$ samples with replacement? In other words, I have a box of balls with $n$ colors. The frequency of each color is given by a function $F(n)$. After sampling the distribution t times with replacement how many of the $n$ colors have been exposed in expectation?
There is an answer here. Using the given solution one could divide the distribution into $n$ discrete points to get the probability of each point and perform the given summation to arrive at the expected number of different colors seen.
Q: Is there a way to go directly from the continuous function $F(n)$ to an approximate or exact solution for the discrete case without the need for the summation given in the above solution?