Joint Probability Distribution Of Number Research Articles

Joint distributions of runs in a sequence of higher-order two-state Markov trials

Consider a time homogeneous {0, 1}-valued m-dependent Markov chain \(\{X_{- m + 1 + n}, n \geqslant 0\}\). In this paper, we study the joint probability distribution of number of 0-runs of length \(k_{0} (k_{0} \geqslant m)\) and number of 1-runs of length \(k_{1} (k_{1} \geqslant m)\) in n trials. We study the joint distributions based on five popular counting schemes of runs. The main tool used to obtain the probability generating function of the joint distribution is the conditional probability generating function method. Further a compact method for the evaluation of exact joint distribution is developed. For higher-order two-state Markov chain, these joint distributions are new in the literature of distributions of run statistics. We use these distributions to derive some waiting time distributions.

Gene survival in the Asian wild horse (Equus przewalskii): I. Dependence of gene survival in the calgary breeding group pedigree

AbstractThe joint probability distribution of the number of distinct (not identical by descent) genes from each founder of the Equus przewalskii population that survive in the five horses of the Calgary Zoological Gardens breeding group has been calculated. The dependence structure of this distribution is investigated, and informative marginal distributions are given, among them the distributions of the genetic contributions of each founder to the Calgary horses and the distribution of wild‐type genes in these horses. The dependence pattern is found to be complex; there is no substitute for exact calculation of the full joint probability distribution of numbers of surviving genes. Probabilities of gene survival give a more complete summary of the genetic structure of a set of individuals than is provided by more routine measures such as heterozygosity or founder contributions. The feasibility of computing these probabilities for small groups of current individuals descended from few founders via long and complex pedigrees, provides a new approach to assessing such groups, and could be used also in selecting animals to form the founder stock of propagules for future reintroduction programs.

Topological cross sections in hadron-nucleon and hadron-nucleus collisions at very high energies

We apply our microscopic model for the topological cross section σ n to produce n charged particles in hadron-proton interactions to hadron-nucleus scattering. The model is based on a stochastic branching process for hadronization. We calculate multiplicity distributions of hadron-nucleus collisions for 50 GeV ⩽ E L ⩽ 400 GeV based on a multiple collision model. The production of “grey” (0.3 < ν c < 0.7) particles is considered together with the shower ( ν c > 0.7) particles in order to test the model for higher number of collisions. The joint probability distribution of numbers of shower and grey particles F( n s, n g) is calculated. Finally, we critically compare the results to experimental data.