Assume that $x_1, ... , x_n$ is a sample of independent observations.
Classically, the degree of belonging $x_i$ into the random sample is equal to $1$, so all observations are weighted equally.
But in case when each observation is (not independently) weighted $(x_1, w_1),...,(x_n, w_n)$ by $w_i \in [0,1]$ a 'weighted data set' is derived. Note that the same $x_i$ values might have different weights.
Is there some formula (or at least a definition) for the distribution of the weigthed data in terms of the distribution of unweighted (equally weighted) data?