The position and gap distribution in stochastic perch system and its application in a model of non-replacing sample

This paper proposes a probability model of stochastic perch system. Picking randomly positions from N positions, the joint probability distribution and the marginal probability distribution of the positions and the gaps are obtained. Furthermore a probability model of sampling without replacement is discussed in this paper, and the conclusion is that the number of experiments in sampling without replacement has the same distribution as the positions in stochastic perch system. Compared with the replacing sample, the conclusion is, when the population is small, the difference between the distributions of sampling without and with replacement is big, but when the population is very large and when the ratio of n to N tend to a constant p, the limit distribution of the number of experiments in sampling without replacement is the same as the distribution of the number of experiments in sampling with replacement.