## Abstract

We consider an r-player version of the famous problem of the points, which was the stimulus for the correspondence between Pascal and Fermat in the 17th century. At each play of a game, exactly one of the players wins a point, player i winning with probability p_{i}. The game ends the first time a player has accumulated his or her required number of points—this requirement being n_{i}for player i. A reliability application would be to suppose that a system is subject to r different types of shocks and failure occurs the first time there have been n_{i}type i shocks for any i = 1, …, r. Our main result is to show that N, the total number of plays, is an increasing failure-rate random variable. In addition, we prove some Schur convexity results regarding P{N ≤ k) as a function of p (for n_{i}≡ n) and as a function of n (for p_{i}≡ 1/r).

Original language | English |
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Pages (from-to) | 663-666 |

Number of pages | 4 |

Journal | Journal of the American Statistical Association |

Volume | 75 |

Issue number | 371 |

DOIs | |

State | Published - Sep 1980 |

Externally published | Yes |

## Keywords

- Duration of play
- Increasing failure rate
- Problem of the points
- Schur convex

## ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty