Timeline for How large can $\mathbf{P}[X_1 + X_2 + X_3 < 2 X_4]$ get?
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| Oct 28 at 18:16 | comment | added | Tobias Fritz | I'm in the process of writing this up as a paper for an expository journal. In case that you're interested in coauthorship @jlewk, just shoot me an email! (Iif not, I'll of course still attribute this idea to you.) | |
| Jul 13 at 8:11 | comment | added | Tobias Fritz | Nice! Applying this method with $Y_{i,j} \sim \frac{1}{2} \delta_0 + \frac{1}{6} \delta_2 + \frac{1}{3} \delta_3$, I actually get a probability of $\approx 0.365$. I will try to push this some further and then write another answer on how I came up with this choice. | |
| Jul 12 at 22:23 | history | edited | Michael Hardy | CC BY-SA 4.0 |
This results in proper spacing to the right of \sim.
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| Jul 12 at 17:30 | history | edited | jlewk | CC BY-SA 4.0 |
typo/aesthetics
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| Jul 12 at 16:33 | comment | added | Will Sawin | In the cases where Tobias Fritz mentioned already finding the optimal value, this procedure (essentially) gets it: It's asymptotically approached here in the limit as $p$ goes to $0$. So it's possible this could be optimal here as well, although I would like to check local optimality also. | |
| Jul 12 at 16:13 | comment | added | jlewk | Thanks. With the explicit definition as a sum, simulations confirm around $3.3-3.4$. | |
| Jul 12 at 16:10 | history | edited | jlewk | CC BY-SA 4.0 |
Add simulations
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| Jul 12 at 15:58 | comment | added | Will Sawin | It might be clearer to definine the distribution all at once as $X_i = Y_{i,1} + t Y_{i,2} + t^2 Y_{i,3} + \dots$ where $Y_{i,j}$ are i.i.d. Bernoulli's with the same probability $p$ and $t$ a very small constant. Then $X_1 + X_2 + X_3 < 2 X_4$ if and only if for some $i$ we have $Y_{1,i} + Y_{2,i} + Y_{3,i} < 2 Y_{4,i}$ and for all $j<i$ we have $Y_{1,j} + Y_{2,j} + Y_{3,j} < 2 Y_{4,j}$. One can derive your $f(p)/ (1-g(p))$ formula from this. | |
| Jul 12 at 15:55 | history | edited | jlewk | CC BY-SA 4.0 |
typo where t is located in the weighted sum
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| Jul 12 at 15:44 | history | answered | jlewk | CC BY-SA 4.0 |