Timeline for Scalar product of random unit vectors
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 29, 2020 at 12:57 | vote | accept | Gin Pat | ||
| May 28, 2020 at 23:39 | answer | added | Iosif Pinelis | timeline score: 8 | |
| May 28, 2020 at 23:35 | comment | added | Dieter Kadelka | I think this is worth an answer. | |
| May 28, 2020 at 22:57 | comment | added | Pierre PC | @DieterKadelka Absolutely, and that's what I actually did. Apologies, a plus sign turned to a negative one; I found $\sqrt{1-x}^{d-3}\mathrm dx$ up to a constant. If $\mathbb S^{d-1}$ is parametrised as $(\sqrt{1-h^2}\cdot\theta,h)$ for $\theta\in\mathbb S^{d-2}$ and $h\in(-1,1)$, then the volume form is $(1-h^2)^{d/2-1}\mathrm d\theta\cdot(1-h^2)^{-1/2}\mathrm dh$. | |
| May 28, 2020 at 22:26 | comment | added | Dieter Kadelka | In additon we may assume that $X \equiv (1,0,\ldots,0)$. | |
| May 28, 2020 at 22:23 | comment | added | Pierre PC | By conditioning with respect to $X$, the solution to both questions is the same. Now the probability that the dot product belongs to $[-1,x]$ is the ratio of the volume of a certain spherical cap to the volume of the whole sphere, which is a simple integral computation. I got a density $(1-x^2)^{(d-1)/2}\mathrm dx$ up to constant, but I might be wrong. There are formulas on Wikipedia. | |
| May 28, 2020 at 22:06 | review | First posts | |||
| May 28, 2020 at 22:35 | |||||
| May 28, 2020 at 22:00 | history | asked | Gin Pat | CC BY-SA 4.0 |