On random split of the segment

Milena Bieniek¹ ; Dominik Szynal²

¹ Department of Economics Maria Curie-Sk/lodowska University Plac Marii Curie-Sk/lodowskiej 5 20-031 Lublin, Poland
² Institute of Mathematics Maria Curie-Sk/lodowska University Plac Marii Curie-Sk/lodowskiej 1 20-031 Lublin, Poland

Applicationes Mathematicae, Tome 32 (2005) no. 3, pp. 243-261

Résumé

We consider a partition of the interval $[0,1]$ by two partition procedures. In the first a chosen piece of $[0,1]$ is split into halves, in the second it is split by uniformly distributed points. Initially, the interval $[0,1]$ is divided either into halves or by a uniformly distributed random variable. Next a piece to be split is chosen either with probability equal to its length or each piece is chosen with equal probability, and then the chosen piece is split by one of the above procedures. These actions are repeated indefinitely. We investigate the probability distribution of the lengths of the consecutive pieces after $n$ splits.

Détail

DOI : 10.4064/am32-3-1

Keywords: consider partition interval partition procedures first chosen piece split halves second split uniformly distributed points initially interval divided either halves uniformly distributed random variable piece split chosen either probability equal its length each piece chosen equal probability chosen piece split above procedures these actions repeated indefinitely investigate probability distribution lengths consecutive pieces after splits