Timeline for Probabilities and rolling 2 dice
Current License: CC BY-SA 2.5
10 events
| when toggle format | what | by | license | comment | |
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| Apr 26, 2010 at 20:54 | comment | added | I. J. Kennedy | The non-rigorous way of seeing the limit is 1/7: Each roll adds an average of 7 to the running total (with a symmetric distribution). "Therefore" about 1/7 of the integers will be landed upon. In other words, a large n has about 1/7 probability of being hit. | |
| Mar 22, 2010 at 15:41 | vote | accept | Stephen Shea | ||
| Mar 15, 2010 at 20:17 | comment | added | Michael Lugo | I did the computation. The maxima are at 16, 26, 34, 43, 51, 60, 69, 78, ..., with a period of approximately 9. In fact the smallest (in modulus) of the $r_k$ are a pair near $.926 \pm .811i$, or $1.231 \exp(.720i)$; thus we expect oscillations with period $2\pi/.720$ or $8.73$ as the contributions from those two roots go in and out of phase with each other. I don't know of a good reason why this number should be larger than 7. | |
| Mar 15, 2010 at 20:06 | comment | added | Stephen Shea | The function seems to have a local max at n=7. I would suppose there is another at 14 (although less pronounced). Is there a point at which this function is monotone? Is there a local max at 7k for all integers k? | |
| Mar 15, 2010 at 20:06 | comment | added | Michael Lugo | Tony, yes it does: if $|t| < 1$ then the sum of the absolute values of the non-constant terms is less than 1. | |
| Mar 15, 2010 at 19:34 | comment | added | Tony Huynh | @Michael. Nice answer. My complex analysis is pretty rusty, but doesn't the triangle inequality imply that $1 / (1-P(z))$ has no singularities inside the unit circle? | |
| Mar 15, 2010 at 19:30 | comment | added | Michael Lugo | Gerhard, I think the "explicit" formula I gave could be used in this way, at least if one explicitly calculated the constants $C_k$ and $r_k$. I don't wish to do this. But I believe that your guess is true. The difference $p(n)-1/7$ decays exponentially fast, so any statement of that form that looks true probably is. | |
| Mar 15, 2010 at 18:58 | comment | added | Gerhard Paseman | Can you use this to say for which n P(n) gets close to 1/7? E.g., is it true that for all n > 21 |P(n) - 1/7| < 1/216? Gerhard "Ask Me About System Design" Paseman, 2010.03.15 | |
| Mar 15, 2010 at 18:54 | comment | added | Michael Lugo | Although I haven't thought about it too hard, this argument should generalize to show that the coefficients of $1/(1-P(z))$, where $P(z)$ is the probability generating function of some positive-integer-valued distribution, approach $1/\mu$ as $n$ gets large where $\mu$ is the mean of the distribution whose pgf is $P(z)$. (That is, $\mu = P^\prime(1)/P(1)$.) The only stumbling block would be showing $1/(1-P(z))$ has no singularities inside the unit circle, which is probably a simple bit of complex analysis. | |
| Mar 15, 2010 at 18:18 | history | answered | Michael Lugo | CC BY-SA 2.5 |