Given a random 2-d sequence $X = (a_1, b_1), \cdots, (a_i, b_i), \cdots, (a_n, b_n)$ on the plane. $X \subset \mathbb{R}^2$. I can compute it's convex hull $CH(X)$ and count the extreme points(points on the boundary).
Suppose $X$ has such property $a_i \ge a_{i+1}$ and $b_i \leq b_{i+1}$. Both $a_i$ and $b_i$ are first generated independently, uniformly random from the same range, and then sorted in order
Now my question is, what's the expected number of extreme points with such a length $n$ sequence?
Here is a plot of sorted random $a$ and $b$:
Concatenate them form $X$ and extreme points of $CH(X)$:
