Finite Population Sampling without Replacement

Personal note of finite population sampling

First moment Population: \( \Omega = \{ x_1, \dots, x_N \} \) Collection of $n$-samples: \(\mathcal{S} = \{ s \in \Omega^n \mid \forall i,j \in s, i \ne j \} \) Collection of $n$-samples containing $x$: \( \mathcal{S}_x = \{ s \in \mathcal{S} \mid x \in s \} \) Observe that \( |\mathcal{S}_x| = \binom{N-1}{n-1} \). Let population mean be zero. $\mu = 0$, i.e. \( \sum_{i = 1}^N x_i = 0 \) [Read More]