# This random variable $Z$ seems to have the same distribution as $\min{X,Y}$ as $|X-Y|$, for $X,Y \sim \operatorname{Unif}(0,1)$.

Based on these (but I hope this is self-contained):

Why does $\min(X,Y)$ and $|X-Y|$ have the same distribution when $X,Y\sim U(0,1)$?

Random points on a circle

For $$X,Y \sim \operatorname{Unif}(0,1)$$, $$\min\{X,Y\}$$ and $$|X-Y|$$ have the same distribution, as explained intuitively in the 1st question. Apparently, for this random variable $$Z$$ described in the 2nd question, $$Z$$ also has the same distribution as $$\min\{X,Y\}$$ and $$|X-Y|$$. $$Z$$ is described as

Suppose 3 (distinct) points are uniformly and independently distributed on a circle of unit length (smaller than a unit circle!). This is really circle and not disc. Call one of these points $$B$$. Let $$Z$$ be the random variable denoting the distance of the point $$B$$ to its neighbour in the anti-clockwise direction.

Of course the precise answer to this is that they all have pdf $$f_Z(z) = 2(1-z)1_{z \in (0,1)}$$. But I'm wondering here if $$Z$$ is actually the absolute difference or minimum or 2 iid uniform(0,1) or if $$Z$$ is something else altogether.

• $X\sim Y$ doesn't implies that $X=Y$.
– Surb
Commented Mar 14, 2021 at 7:56
• @Surb yeah I know it's like 2 bernoulli's with the same probability parameter $p$ but different spaces. or even the same space like $(0,1)$ but different intervals each like $1_{(0,0.4)}$ or $1_{(0.2,0.6)}$. Just wanted to check here if perhaps $Z$ was like a $\min$ or an absolute value or just really something else.
– BCLC
Commented Mar 14, 2021 at 8:01

I believe $$Z=\min\{X,Y\}$$, where $$X$$ and $$Y$$ are described in David K's answer:
Let $$X$$ be the anticlockwise distance along the circle's circumference from $$B$$ to $$A$$. Let $$Y$$ be the anticlockwise distance from $$B$$ to $$C$$.
Since the three points are independently uniformly distributed along the circumference of the circle, $$X$$ and $$Y$$ are iid variables with uniform distributions on $$[0,1).$$