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| HAL : hal-00506492, version 2 |
| arXiv : 1007.4870 |
| Fiche détaillée |  Récupérer au format |
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| Versions disponibles : | v1 (28-07-2010) | v2 (30-07-2010) |
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| (Very) short proof of Rayleigh's Theorem (and extensions) |
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| Olivier Bernardi 1 |
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| (28/07/2010) |
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| Consider a walk in the plane made of $n$ steps of length 1, with directions chosen independently and uniformly at random at each step. Rayleigh's Theorem asserts that the probability for such a walk to end at a distance less than 1 from its starting point is $1/(n+1)$. We give an elementary proof of this result. We also prove the following generalization valid for any probability distribution $\mu$ on the positive real numbers: if two walkers start at the same point and make respectively $i$ and $j$ independent steps with uniformly random directions and with lengths chosen according to $\mu$, then the probability that the first walker ends farther than the second is~$i/(i+j)$. |
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| 1 : | Massachusets Institute of Technology (MIT) |
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MIT |
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| Domaine | : | Mathématiques/Combinatoire Mathématiques/Probabilités |
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| Liste des fichiers attachés à ce document : | ||||||||||
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| hal-00506492, version 2 | |
| http://hal.archives-ouvertes.fr/hal-00506492 | |
| oai:hal.archives-ouvertes.fr:hal-00506492 | |
| Contributeur : Olivier Bernardi | |
| Soumis le : Vendredi 30 Juillet 2010, 15:27:52 | |
| Dernière modification le : Vendredi 30 Juillet 2010, 15:39:23 | |
