On multivariate sampling with and without replacement

In the chapters I - VII we have succeeded in studying, the most important theoretical properties of probability distributions which arises in common sampling with and without replacement. For Univariate Hypergeometric sampling, we have shown that the different functions obey; a Factorial Binomial law, and proceeding from this we were able to study by similar methods the generalisations of Binomial and hypergeometric (or Factorial Binomial) sampling to more variates and rmultinomial cases.All these distributions have very simple p.q.f.'s and we have, therefore, use the generating, functions as starting point in the study. Simple symbolic forms could always be written down for the probability functions.We have obtained the Multivariate Normal correlation distribution rot, only as a a limiting form of the Multivariate Multinormal, but also, as a limiting form of the Multivariate Factorial Multinomial in cases where the latter does not first tend to the Multinomial. This is the essential factor behind the practical importance of the Normal correlation distribution, since in actual practice we always have sampling without, replacement from a limited population