Th birthday problem (or paradox) has been done in many way, with around a dozen thread only on math.stackexchange. The way it is expressed is usually the following:
"Let us take $n$ people "independently" (no twins, etc.). What is the probability that no two people share the same birthday?"
It is abstracted in the following way:
"Let $X_1$, $\cdots$, $X_n$ be $n$ i.i.d. random variables taken uniformly in $[[1, 365]]$. What is the probability that all the $X_i$'s are distinct?"
There are many generalizations, for instance:
"Let $n$, $d$ be two positive integers, $n \leq d$. Let $X_1$, $\cdots$, $X_n$ be $n$ i.i.d. random variables taken uniformly in $[[1, d]]$. What is the probability $p(n,d)$ that all the $X_i$'s are distinct?"
One can show that in the regimen $1 \ll n \ll d$, the probability $p(n,d)$ is logarithmically equivalent to something like $e^{-\frac{n^2}{2d}}$ (Wikipedia) or $e^{-\frac{n^2}{d}}$ (my computations)*. This problem can be reduced to simple combinatorics, and Stirling's formula (for instance) gives the solution.
However, in the real world, the birthdays are not distributed that way. One might also want to estimate the probability that two peoples are born the same half-day, the same hour, etc. The following generalization seems natural:
"Let $\mu$ be a probability measure on $[0,1]$ absolutely continuous with respect to the Lebesgue measure. Let $n$, $d$ be two positive integers. For $k \in [[0,d-1]]$, let $a_k := [k/d, (k+1)/d]$. Let $X_1$, $\cdots$, $X_n$ be $n$ i.i.d. random variables in $[0,1]$ with distribution $\mu$. What is the probability $p(n,d)$ that all the $X_i$'s lie in different elements of the partition?"
I would expect the solution to be something like $e^{-C(\mu) \frac{n^2}{d}}$, with perhaps some explicit expression of $C(\mu)$. But the combinatorial solutions do not work as well in this setting, and all I can get are very crude bounds when the density of $\mu$ is bounded. In addition, I would expect $C(\mu)$ to be minimal when $\mu$ is the Lebesgue measure, but I don't know how to prove it. One might wonder what happens when $\mu$ is no longer absolutely continuous, but this might be a bit too broad of a generalization.
I am sure this problem has been done to death, but I don't have any access to the literature right now, and quick search didn't yield anything (the generalizations of the birthday problem I found are quite different). Any result/proof/reference related to the problems above would be nice.
.* By the way, any rigorous proof of either of the two facts (or of any similar-sounding result) is appreciated. I don't know which I can trust more, between my computations and Wikipedia.