Sampling Distribution

A Sampling Distribution is the probability that a distribution of a statistic is obtained through repeated sampling of a specific population.

Suppose $X_i$ is independent and follows $\text{Normal}(\mu ,{\sigma }^2)$

$\bar{X}\approx \text{Normal}(\mu ,\frac{\sigma }{\sqrt{n}})$

Expected Value

$$\mathbb{E}[\bar{X}]=\mu$$

Variance

$$\mathrm{Var}(\bar{X})=\frac{{\sigma }^2}{n}$$