Expectation of minimum of a random sample

Question

Suppose we have $N$ points uniformly sampled on $0<x<1$. Let $0<S<1$ and $N^\ast=\lfloor{NS}\rfloor$, and we randomly draw $N^\ast$ points from the $N$ points (without return). What is the distribution of the minimum value of the $N^\ast$ points? Intuitively, the expectation decreases with $S$. I thought about this for quite a while but got no clue. A simulation using $N=10000$ gives a distribution like this plot. Looks like that the region is bounded by two exponential functions.

Answer 1

What is the probability that the minimum is greater than some fixed number $y.$ Well, this is only true if (and only if) all your independent uniform random variables are greater than $y,$ the probability of which is $(1-y)^{NS}$ (in your notation). So, the CDF of your minimum is $1-(1-y)^{NS},$ and so the density function is $NS(1-y)^{NS-1}.$