Timeline for Random process on a sequence of rolls of an $n$-sided die
Current License: CC BY-SA 4.0
15 events
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| Dec 17, 2020 at 13:12 | comment | added | Let101 | I was also wondering how does this problem change if, whenever $X\le k$, we decrease $t$ by $x+1$ (instead of by $1$), and the stopping condition is $t\le 0$? I guess in this case the "symmetry" of the problem could make it easier to solve. | |
| Dec 17, 2020 at 13:10 | comment | added | Let101 | Thank you @NateEldredge for your comments. I see. Let $T$ be the random variable equal to total number of rolls (taking value $\tau\in[1,\infty]$), and let $D(r)~~ (:=D_{k,n}(r))$ be the total number of dollars won at round $r$. $D(r)<0$ means that we are loosing money at round $r$. Instead of asking for $\mathbb{E}[D(T)]$ as in the original problem version, would it be more meaningful to ask for $\mathbb{E}[\lim\sup_{r\to T}D(r)]$ and $\mathbb{E}[\lim\inf_{r\to T}D(r)]$? | |
| Dec 17, 2020 at 4:35 | comment | added | Nate Eldredge | Wald does seem helpful because this is a random walk $S_m$ stopped at a stopping time $\tau$. However to apply it, we need to have $E[\tau] < \infty$ and as I note above, in most cases we do not even have $\tau < \infty$ a.s. | |
| Dec 17, 2020 at 4:32 | comment | added | Nate Eldredge | If I'm not mistaken, whenever we take $k < n-1$, the counter $t$ has positive drift and there is a positive probability that the game never terminates. What do we mean by "total number of dollars won" in this case? | |
| Dec 16, 2020 at 23:15 | history | edited | Let101 | CC BY-SA 4.0 |
added 1 character in body
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| Dec 16, 2020 at 22:44 | history | edited | Wlod AA | CC BY-SA 4.0 |
Question named as such
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| Dec 16, 2020 at 22:40 | comment | added | Let101 | Is Wald's equation useful in this case? | |
| Dec 16, 2020 at 22:40 | history | edited | Let101 | CC BY-SA 4.0 |
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| Dec 16, 2020 at 22:28 | comment | added | Wlod AA | I feel that the q. about Wald should be moved to a comment. | |
| Dec 16, 2020 at 21:42 | history | edited | Let101 | CC BY-SA 4.0 |
added 9 characters in body
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| Dec 16, 2020 at 21:42 | comment | added | Let101 | Yes @kodlu . The meaning basically is "When is convenient to play this game? (without loosing time)" Edited. | |
| Dec 16, 2020 at 21:37 | comment | added | kodlu | Is "larger than" a strict inequality? | |
| Dec 16, 2020 at 20:04 | history | edited | Wlod AA | CC BY-SA 4.0 |
a further improvement
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| Dec 16, 2020 at 19:59 | history | edited | Wlod AA | CC BY-SA 4.0 |
a clearer (more precise) formulation + fixing one or two typos.
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| Dec 16, 2020 at 19:30 | history | asked | Let101 | CC BY-SA 4.0 |