Notable relations

Author

Affiliation

Beniamino Sartini

University of Bologna

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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.  $$\begin{array}{}\text{(1)}& {\chi }_{\nu }^2\sim Z_1^2+\cdots +Z_i^2+\cdots +Z_{\nu }^2\text{,}\end{array}$$ 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.  $$\begin{array}{}\text{(2)}& t(\nu )\sim \frac{Z}{\sqrt{\frac{{\chi }_{\nu }^2}{\nu }}}=\frac{\sqrt{\nu }X}{\sqrt{{\chi }_{\nu }^2}}\end{array}$$ 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.
$$\begin{array}{}\text{(3)}& F_{{\nu }_1,{\nu }_2}\sim \frac{\frac{{\chi }_{{\nu }_1}^2}{{\nu }_1}}{\frac{{\chi }_{\nu }^2}{{\nu }_2}}=\frac{{\nu }_2}{{\nu }_1}\frac{{\chi }_{{\nu }_1}^2}{{\chi }_{{\nu }_2}^2}\end{array}$$