## 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]