On the waiting time in a generalized roulette game
type: Item
instance of: scholarly article
title: On the waiting time in a generalized roulette game
zbMATH Open document ID: 0549.60039
DOI: 10.1016/0167-7152(84)90021-X
published in: Statistics \& Probability Letters
publication date: 1984-01-01T00:00:00Z
review text: This paper considers the following problem: a roulette has numbers 0,1,2,...,m-1 as possible outcomes; the game continues until the first time that numbers 1,2,...,k have all appeared before 0 appears; if a zero appears too soon, the game starts again from scratch. A generalization of this problem considers a multinomial experiment with outcomes \(E_ 0,E_ 1,...,E_{N+1}\); the experiment is repeated until \(E_ 1,...,E_ N\) have all occurred before \(E_ 0\) occurs. Let S be the total time elapsed until the game ends; expressions for the expectation and the variance, as well as for the moment generating function of S are obtained. Also, some asymptotic results are derived for S as N tends to infinity: in the particular case where \(E_ 0,...,E_ N\) have all the same probability \(1/(N+1)\), and \(E_{N+1}\) has probability 0, for example, \(S/(N+1)^ 2\) converges in distribution to an exponential random variable with parameter 1. The authors then go back to the original roulette problem, and to the problem of rolling two six-sided dice repeatedly, and determining the number of rolls necessary to obtain 4,5,6,8,9 and 10 before 7 is obtained, and apply their general formulas to these two particular cases.
Mathematics Subject Classification ID: 60G40
Mathematics Subject Classification ID: 60C05
Mathematics Subject Classification ID: 60G50
zbMATH DE Number: 3876314
zbMATH Keywords: stopping time
zbMATH Keywords: stopped sum
zbMATH Keywords: recurrent event
zbMATH Keywords: Poisson process
zbMATH Keywords: waiting time until success
zbMATH Keywords: roulette
zbMATH Keywords: moment generating function
author: Allan Gut
author: Lars Holst
reviewed by: J. Gianini Pettitt
MaRDI profile type: Publication
cites work: Q3311389
cites work: On sequential occupancy problems
cites work: Q4114574
cites work: Repeated Independent Trials and a Class of Dice Problems
description: scientific article; zbMATH DE number 3876314