Suppose an experiment has $N$ possible outcomes, each with probability $1/N$. If I repeat the experiment $a N$ times where $a$ is a whole number, what is the probability that every single one of the $N$ possible outcomes is achieved over the course of this repetition? What is the answer for a more general probability distribution?
Basic probability question on repetition of an experiment
1
$\begingroup$
probability
probability-distributions
-
0I think this has something to do with the pigeonhole principle but I can't quite wrap my finger around it. This came up when I was trying to find a good way to get out of local maxima for an MCMC algorithm. – 2017-02-14
-
1This is similar to the [coupon collector's problem](https://en.wikipedia.org/wiki/Coupon_collector's_problem) but is instead asking for the probability distribution as opposed to simply the expected number. As for an approach I recommend using [inclusion-exclusion](https://en.wikipedia.org/wiki/Inclusion%E2%80%93exclusion_principle) or using generating functions as in @Marko's answer [here](http://math.stackexchange.com/a/2141697/179297). Also, no this isn't related to pigeonhole principle. – 2017-02-14
1 Answers
1
The answer to your specific question is
$$\sum_{M=0}^N (-1)^M {N\choose M}\left(1-\frac{M}{N}\right)^{aN} = \dfrac{N! \,}{N^{aN}} S_2(aN,N)$$
where $S_2(n,m)$ is a Stirling number of the second kind, sometimes written ${ n\brace m}$
-
0(+1) (... which is a special case of your answer [
](http://math.stackexchange.com/a/32816/16397)) – 2017-02-15