Moments for generalizations of a coin flip game
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arXiv
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866911698527453184 |
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| author | Huang, Jia |
| author_facet | Huang, Jia |
| contents | We derive a recursive formula for the moments of the number of flips using a possibly biased coin to produce a prescribed finite binary string $S$ when $S$ is either a run of heads or a run of heads followed by a tails. Our recursive formula involve certain sums, which we simplify by using a one-parameter extension of the well-studied Eulerian number, which belongs to the two-parameter family of numbers introduced by Graham, Knuth, and Patashnik. We also use the Goulden--Jackson cluster method and Faà di Bruno's formula to establish a closed formula for the moments in a more general situation where a die having an arbitrary number of faces with possibly different probabilities is rolled repeatedly until a prescribed finite word occurs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_19904 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Moments for generalizations of a coin flip game Huang, Jia Combinatorics Probability 05A15, 11B39 We derive a recursive formula for the moments of the number of flips using a possibly biased coin to produce a prescribed finite binary string $S$ when $S$ is either a run of heads or a run of heads followed by a tails. Our recursive formula involve certain sums, which we simplify by using a one-parameter extension of the well-studied Eulerian number, which belongs to the two-parameter family of numbers introduced by Graham, Knuth, and Patashnik. We also use the Goulden--Jackson cluster method and Faà di Bruno's formula to establish a closed formula for the moments in a more general situation where a die having an arbitrary number of faces with possibly different probabilities is rolled repeatedly until a prescribed finite word occurs. |
| title | Moments for generalizations of a coin flip game |
| topic | Combinatorics Probability 05A15, 11B39 |
| url | https://arxiv.org/abs/2605.19904 |