Sampling Distributions

Distributions Related to the Normal

• If $X \sim \mathcal{N}(\mu, \sigma^2)$ then $\frac{X - \mu}{\sigma} \sim \mathcal{N}(0, 1)$ and $\big(\frac{X - \mu}{\sigma}\big)^2 \sim \chi^2_1$.
• Independent $\chi^2$'s add, $\chi^2_i + \chi^2_j \sim \chi^2_{i+j}$, so the sum of $n$ independent $\chi^2_1$ is $\chi^2_n$.
• $\chi^2_k = \mathrm{Gamma}\big(\alpha = \frac{k}{2}, \lambda = \frac{1}{2}\big)$.
• If $Z \sim \mathcal{N}(0, 1)$ and $U \sim \chi^2_n$ are independent, $\frac{Z}{\sqrt{U/n}} \sim t_n$.
• If $U \sim \chi^2_m$ and $V \sim \chi^2_n$ are independent, $\frac{U/m}{V/n} \sim F_{m,n}$. In particular $T^2 \sim F_{1,n}$ for $T \sim t_n$.

Independence is needed for all of these.

Sample Mean and Variance

For $X_1, \ldots, X_n$ IID with $E[X_i] = \mu$, $\mathrm{Var}(X_i) = \sigma^2$,

$$\bar{X} = \frac{1}{n}\sum_{i=1}^n X_i, \quad E[\bar{X}] = \mu, \quad \mathrm{Var}(\bar{X}) = \frac{\sigma^2}{n}, \qquad S^2 = \frac{1}{n - 1}\sum_{i=1}^n (X_i - \bar{X})^2, \quad E[S^2] = \sigma^2$$

If moreover $X_i \sim \mathcal{N}(\mu, \sigma^2)$:

• $\bar{X}$ and $(X_1 - \bar{X}, \ldots, X_n - \bar{X})$ are independent, hence $\bar{X}$ and $S^2$ are independent,
• $\frac{(n - 1)S^2}{\sigma^2} \sim \chi^2_{n-1}$,
• $\frac{\bar{X} - \mu}{S / \sqrt{n}} \sim t_{n-1}$.