Timeline for Geometry interpretation of any continuous random variable
Current License: CC BY-SA 4.0
20 events
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| Jan 21, 2021 at 16:07 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Jan 21, 2021 at 15:34 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Jan 21, 2021 at 15:22 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Jan 21, 2021 at 15:18 | comment | added | Iosif Pinelis | @PaataIvanishvili : Thank you for this comment. I have now added a corresponding comment to the above answer. | |
| Jan 21, 2021 at 15:16 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Jan 21, 2021 at 1:18 | comment | added | Paata Ivanishvili | @IosifPinelis, "... is exactly 𝜇. I now doubt that". If $d\mu(x) = p(x)dx$ with bounded $p$, then I think Kelleler has a theorem (see discussion on top of page 1784 projecteuclid.org/download/pdf_1/euclid.aop/1176990236) that if $0\leq f \leq 1$ is measurable on $\mathbb{R}^{n}$, then there exists $A \subset \mathbb{R}^{n}$ such that $f(x)dx$ and $1_{A}(x)dx$ have the same (1-dimensional) marginals. Now apply the Kelleler's theorem to $f(x) = \frac{1}{\|p\|_{\infty}^{n}}\prod_{1}^{n} p(x_i)$. | |
| Jan 20, 2021 at 22:06 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Jan 20, 2021 at 17:24 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Jan 20, 2021 at 17:15 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Jan 20, 2021 at 17:03 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Jan 20, 2021 at 16:56 | comment | added | Iosif Pinelis | @RyanChen : Thank you for your comment. I have quite substantially edited the answer. If $\mu$ has a smooth bounded density $p$ such that $p(t)>0$ for all real $t$ (which can be assumed), then the corresponding regions $A^\epsilon$ will be simply connected. If you want to explore this topological question further, you may want to post it separately. | |
| Jan 20, 2021 at 16:35 | comment | added | Iosif Pinelis | @WillSawin : Thank you for your comment. | |
| Jan 20, 2021 at 16:34 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Jan 20, 2021 at 6:08 | comment | added | RyanChan | @losifPinelis Very useful analysis. If we pose some topologic constraint on the desired geometric region (such as a simply connected region), I wonder whether the results above developed for finite $n$ still hold. | |
| Jan 20, 2021 at 5:19 | vote | accept | RyanChan | ||
| Jan 20, 2021 at 5:18 | vote | accept | RyanChan | ||
| Jan 20, 2021 at 5:18 | |||||
| Jan 20, 2021 at 2:32 | comment | added | Will Sawin | Nice approach! Any marginal of the uniform distribution of a bounded set with positive Lebesgue measure will have bounded probability density function, so maybe this works for all measures with bounded pdf... | |
| Jan 19, 2021 at 19:09 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Jan 19, 2021 at 18:23 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Jan 19, 2021 at 18:14 | history | answered | Iosif Pinelis | CC BY-SA 4.0 |