Communications in Information and Systems

Volume 12 (2012)

Number 4

Random walk and linear switching systems

Pages: 277 – 299

DOI: http://dx.doi.org/10.4310/CIS.2012.v12.n4.a3

Authors

Yulei Pang (Department of Mathematics and Statistics, Texas Tech University, Lubbock, Texas, U.S.A.)

Alex Wang (Department of Mathematics and Statistics, Texas Tech University, Lubbock, Texas, U.S.A.)

Xiaozhen Xue (Department of Mathematics and Statistics, Texas Tech University, Lubbock, Texas, U.S.A.)

Clyde F. Martin (Department of Mathematics and Statistics, Texas Tech University, Lubbock, Texas, U.S.A.)

Abstract

In this paper we address the question “for a deck of cards, how many times a top-in shuffle should be performed before the top card goes back to the original position?” This problem has been studied in the literature but we are interested in the implications for linear switching systems. We simulate top-in shuffling for 6, 12, and 54 cards, and determine the underlying statistics. Finally we prove that the distribution of the stoping time is an exponential distribution, and the expect value approaches to that of the uniform distribution for large number of shuffling. We make essential use of the properties of linear, stochastic switching systems.

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