limited due to the strong assumption about the y observations. This has led to the development of alternative, more general, estimators. Survey sampling perspective From a model based perspective, we are interested in estimating the variance of the weighted mean when the different are not i.i.d random variables. An alternative perspective for this problem is that of some arbitrary sampling design of the data in which units are selected with unequal probabilities (with replacement). In Survey methodology, the population mean, of some quantity of interest y, is calculated by taking an estimation of the total of y over all elements in the population (Y or sometimes T) and dividing it by the population size – either known () or estimated (). In this context, each value of y is considered constant, and the variability comes from the selection procedure. This in contrast to "model based" approaches in which the randomness is often described in the y values. The survey sampling procedure yields a series of Bernoulli indicator values () that get 1 if some observation i is in the sample and 0 if it was not selected. This can occur with fixed sample size, or varied sample size sampling (e.g.: Poisson sampling). The probability of some element to be chosen, given a sample, is denoted as , and the one-draw probability of selection is (If N is very large and each is very small). For the following derivation we'll assume that the probability of selecting each element is fully represented by these probabilities. I.e.: selecting some element will not influence the probability of drawing another element (this doesn't apply for things such as cluster sampling design). Since each element () is fixed, and the randomness comes from it being included in the sample or not (), we often talk about the multiplication of the two, which is a random variable. To avoid confusion in the following section, let's call this term: . With the following expectancy: ; and variance: . When each element of the sample is inflated by the inverse of its selection probability, it is termed the -expanded y values, i.e.: . A related quantity is -expanded y values: . As above, we can add a tick mark if multiplying by the indicator function. I.e.: In this design based perspective, the weights, used in the numerator of the weighted mean, are obtained from taking the inverse of the