Some Useful Moment Results in Sampling Problems

We consider the standard sampling problem involving a finite population of N objects and a sample of n objects taken from this population using simple random sampling without replacement. We consider the relationship between the moments of the sampled and unsampled parts and show how these are related to the population moments. We derive expectation, variance, and covariance results for the various quantities under consideration and use these to obtain standard sampling results with an extension to variance estimation with a finite population correction. This clarifies and extends standard results in sampling theory for the estimation of the mean and variance of a population.