Written By: Qingyang Xu (website)

Date Created: June 14, 2022

Last Modified: August 7, 2023

Chapter summary of “Mathematical Statistics and Data Analysis” (by John Rice)

1. Distributions derived from normal distribution

1.1 Definition

Let $Z_i \sim N(0,1)$ denote IID standard normal RV.

$\chi^2$-squared distribution with n d.o.f.: $\chi^2_n=\sum_{i=1}^n Z_i^2$

• Density $f(v) \propto v^{n/2-1} e^{-v/2}$
• Intuition: model residual sum of squared error

$t$-distribution with n d.o.f. $t_n= \frac{Z_0}{\sqrt{\chi_n^2/n}}$ (e.g., OLS $\hat{\beta}$) where $Z_0$ and $\chi^2_n$ are independent

• Density $f(t）\propto (1+t^2/n)^{-(n+1)/2} \rightarrow e^{-t^2/2}$ as $n \rightarrow \infty$
• Intuition: $t=(\hat{\beta}-\beta^0)/\sqrt{s^2}$ for single hypothesis test

$F$-distribution with (m,n) d.o.f. $F_{m,n}=\frac{\chi^2_m/m}{\chi^2_n/n}$ (e.g., ANOVA) where $\chi^2_m$ and $\chi^2_n$ are independent

• Density $f(w) \propto w^{m/2-1} (1+wm/n)^{-(m+n)/2}$