Timeline for Does taking minimum preserve density monotonicity?
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
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| Apr 5, 2022 at 18:29 | comment | added | Nikolay | Thanks for this great counterexample! | |
| Apr 5, 2022 at 18:27 | vote | accept | Nikolay | ||
| Apr 5, 2022 at 14:55 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Apr 5, 2022 at 13:45 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Apr 5, 2022 at 8:12 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Apr 5, 2022 at 8:06 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Apr 5, 2022 at 7:50 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Apr 5, 2022 at 7:25 | comment | added | Iosif Pinelis | @Nikolay : Sorry for the mistake. Now I have another counterexample. | |
| Apr 5, 2022 at 7:24 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Apr 5, 2022 at 6:36 | comment | added | Nikolay | The implied density function is $(2-3t/2)1\{0<t\le 1/3\}+$ $1\{1/3<t\le 2/3\}+$ $3/2(1-t)1\{2/3<t<1\}$ which is non-increasing. | |
| Apr 5, 2022 at 6:22 | comment | added | Nikolay | Thanks a lot for your answer! I agree that $X$ and $Y$ are uniformly distributed on [0,1] but I think that there is a mistake in the density of the minimum. For example, consider $\Pr(\min(X,Y)\le 1/3)$. This probability is the probability of 3 dark-brown squares plus the probability of 2 light-brown squares. This gives us 7/12. On the other hand, from your graph this probability is 1/6. In general, I calculated for your example that $\Pr(\min\{X,Y\}\le t)$ equals $2t-3t^2/4$ for $t\in (0, 1/3]$, $1/4+t$ for $t\in(1/3, 2/3]$ and $1/4+3t/2 - 3t^2/4$ for $t\in(2/3, 1)$ | |
| Apr 5, 2022 at 4:42 | history | answered | Iosif Pinelis | CC BY-SA 4.0 |