Notable relations

Author

Affiliation

Beniamino Sartini

University of Bologna

library(ggplot2)
library(tidyverse)
library(backports)
library(latex2exp)
set.seed(1)

1 Chi squared

The chi2 distribution (\(\chi^2\)) with \(\nu\) degrees of freedom, namely \(\chi^2(\nu)\), is defined as the sum of \(\nu\)-independent and identically distributed standard normal random variables, i.e.  \[ \chi^{2}_{\nu} \sim Z_1^2 + \dots + Z_i^2 + \dots + Z_{\nu}^2 \text{,} \tag{1}\] where for \(i = 1, \dots, \nu\), the notation \(Z_i \sim N(0,1)\) denote a standard normal random variable.

2 Student-t

The Student-t distribution with \(\nu\) degrees of freedom, namely \(t(\nu)\), is defined as the ratio of two independent random variables. In specific, a standard normal random variable and the square root of a \(\chi^2(\nu)\) divided by its degrees of freedom \(\nu\), i.e.  \[ t(\nu) \sim \frac{Z}{\sqrt{\frac{\chi^{2}_{\nu}}{\nu}}} = \frac{\sqrt{\nu}X}{\sqrt{\chi^{2}_{\nu}}} \tag{2}\] where the notation \(Z \sim N(0,1)\) denote a standard normal random variable.

3 Fisher–Snedecor

The Fisher–Snedecor distribution with \(\nu_1\) and \(\nu_2\) degrees of freedom, often denoted as F, is defined as the ratio of two independent chi2 random variables, each one divided by its degrees of freedom, i.e.
\[ F_{\nu_1, \nu_2} \sim \frac{\frac{\chi^{2}_{\nu_1}}{\nu_1}}{\frac{\chi^{2}_{\nu}}{\nu_2}} = \frac{\nu_2}{\nu_1}\frac{\chi^{2}_{\nu_1}}{\chi^{2}_{\nu_2}} \tag{3}\]