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When we work with weighted samples, it’s essential to introduce adjustments for the sample size. Indeed, let’s consider two following weighted samples: \[\mathbf{x}_1 = \{ x_1, x_2, \ldots, x_n \}, \quad \mathbf{w}_1 = \{ w_1, w_2, \ldots, w_n \}, \] \[\mathbf{x}_2 = \{ x_1, x_2, \ldots, x_n, x_{n+1} \}, \quad \mathbf{w}_2 = \{ w_1, w_2, \ldots, w_n, 0 \}. \] Since the weight of \(x_{n+1}\) in the second sample is zero, it’s natural to expect that both samples have the same set of properties. Ho...