Sampling problems for randomly broken sticks
Thierry Huillet · Journal of Physics A Mathematical and General · 2003
Consider the random partitioning model of a population (represented by a stick of length 1) into n species (fragments) with identically distributed random weights (sizes). Upon ranking the fragments' weights according to ascending sizes, let S m : n be the size of the m th smallest fragment. Assume that some observer is sampling such populations as follows: drop at random k points (the sample size) onto this stick and record the corresponding numbers of visited fragments. We shall investigate the following sampling problems: (1) what is the sample size if the sampling is carried out until the first visit of the smallest fragment (size S 1: n )? (2) For a given sample size, have all the fragments of the stick been visited at least once or not? This question is related to Feller's random coupon collector problem. (3) In what order are new fragments being discovered and what is the random number of samples separating the discovery of consecutive new fragments until exhaustion of the list? For this problem, the distribution of the size-biased permutation of the species' weights, as the sequence of their weights in their order of appearance is needed and studied.