Chi-Squared Distribution (χ²₍k₎) ○◂｜appendage｜___｜20260628155835-01-•

Chi-squared	…	…	…
Probability density function	…	…	…
Cumulative distribution function	…	…	…
Notation	${\chi }^2(k)$ or ${\chi }_k^2$	…	…
Parameters	$k\in {\mathbb{N}}^{\ast }$ (known as “degrees of freedom”)	…	…
Support	$x\in (0,+\infty )$	…	…
PDF	$\frac{1}{2^{k/2}\Gamma (k/2)}{x}^{(k/2)-1}{e}^{-x/2}$	…	…
CDF	$\frac{1}{\Gamma (k/2)}\gamma \left(\frac{k}{2},\frac{x}{2}\right)$	…	…
Mean	$k$	…	…
Median	$\approx k(1-\frac{2}{9k}{)}^3$	…	…
Mode	$\max (k-2,0)$	…	…
Variance	$2k$	…	…
Skewness	$\sqrt{8/k}$	…	…
Excess kurtosis	$\frac{12}{k}$	…	…
Entropy	$\begin{array}{rl}\frac{k}{2}& +\log \left(2\Gamma \left(\frac{k}{2}\right)\right)\\ & +\left(1-\frac{k}{2}\right)\psi \left(\frac{k}{2}\right)\end{array}$	…	…
MGF	$(1-2t{)}^{-k/2}$ for $t<\frac{1}{2}$	…	…
CF	$(1-2it{)}^{-k/2}$1	…	…
PGF	$(1-2\ln t{)}^{-k/2}$ for $0<t<\sqrt{e}$	…	…

(echo:: @ ᯤ)

Footnotes

1. M.A. Sanders. “Characteristic function of the central chi-square distribution” (PDF). Archived from the original (PDF) on 2011-07-15. Retrieved 2009-03-06. ↩