4
$\begingroup$

Suppose we have a semicircle with radius $1$: We choose two random points with a uniform distribution, that is, that if we pick random points inside it, they will be uniformly distributed. The supremum of the distance between those two points is $2$.

  • What is the probability density function $\operatorname{f}\left(x\right)$ of the distribution of the distances ?. That is, the function such that $\int_{0}^{y}\operatorname{f}\left(x\right)\,{\rm d}x$ gives you the probability that the distance between two randomly selected poins is less than $y$.

Two random points picked in the semicircle

Even without finding this function, what is the average distance between two randomly selected points inside the semicircle ?.

$\endgroup$
4
  • 2
    $\begingroup$ After performing extensive computations with the aid of Mathematica, I obtained the average distance as: $$\frac{256}{45\pi }-\frac{1472}{135\pi^2}\approx 0.706053$$ I ran a Monte-Carlo simulation using $10^7$ samples and the resulting estimate was in good agreement with the above analytic result. $\endgroup$
    – Sangchul Lee
    Commented Jun 3 at 0:17
  • $\begingroup$ @SangchulLee I remember -sometimes ago- I was trying to do that calculation, i.e. $\operatorname{f}\left(x\right)$. It was hard and I never completed it. $\endgroup$
    – Felix Marin
    Commented Jun 3 at 3:57
  • 1
    $\begingroup$ @FelixMarin Without the heavy aid of Mathematica, I would also have never completed it by myself. The intermediate formulas were so nasty that I didn't believe it has a closed form. $\endgroup$
    – Sangchul Lee
    Commented Jun 3 at 4:23
  • $\begingroup$ A Desmos simulation: desmos.com/calculator/uraefstdwb $\endgroup$
    – ploosu2
    Commented Jun 3 at 8:15

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .