  Multivariate hypergeometric distribution arises frequently in elementary
statistics and probability courses, for simultaneously studying the occurence
law of specified events, when sampling without replacement from a finite
population with fixed number of classification. Covariance matrix of this
distribution is well known to be identical to its multinomial counterpart
multiplied by 1-(n-1)/(N-1), with N and n being population and sample sizes,
respectively. It appears to however, have been less discussed in the literature
about the meaning of this relationship, especially regarding the specific form
of the multiplier. Based on an augmenting argument together with probabilistic
symmetry, we present a more transparent understanding for the covariance
structure of the multivariate hypergeometric distribution. We discuss
implications of these combined techniques and provide a unified description
about the relative efficiency for estimating population mean based on simple
random sampling, probability proportional-to-size sampling and adaptive cluster
sampling, with versus without replacement. We also provide insight into the
classic random group method for variance estimation.
