User:Minzastro/sandbox


Chi-squared distribution	Normal distribution	Log-normal	Student's t	Laplace distribution	Weibull distribution	Pareto distribution	chi-squared
parameters	$\mu \in \mathbb {R}$ — mean (location) $\sigma ^{2}>0$ — variance (squared scale)	$\mu \in (-\infty ,+\infty )$ , $\sigma >0$	$\nu >0$ degrees of freedom (real)	$\mu$ location (real) $b>0$ scale (real)	$\lambda \in (0,+\infty )\,$ scale $k\in (0,+\infty )\,$ shape	x_m > 0 scale (real) α > 0 shape (real)	$k\in \mathbb {N} _{>0}~~$ (known as "degrees of freedom")
pdf_image	The red curve is the standard normal distribution	Some log-normal density functions with identical parameter $\mu$ but differing parameters $\sigma$				Pareto Type I probability density functions for various α with x_m = 1. As α → ∞ the distribution approaches δ(x − x_m) where δ is the Dirac delta function.
pdf	${\frac {1}{\sqrt {2\pi \sigma ^{2}}}}\,e^{-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}}$	${\frac {1}{x\sigma {\sqrt {2\pi }}}}\ e^{-{\frac {\left(\ln x-\mu \right)^{2}}{2\sigma ^{2}}}}$	$\textstyle {\frac {\Gamma \left({\frac {\nu +1}{2}}\right)}{{\sqrt {\nu \pi }}\,\Gamma \left({\frac {\nu }{2}}\right)}}\left(1+{\frac {x^{2}}{\nu }}\right)^{-{\frac {\nu +1}{2}}}\!$	${\frac {1}{2\,b}}\exp \left(-{\frac {\|x-\mu \|}{b}}\right)\,$	$f(x)={\begin{cases}{\frac {k}{\lambda }}\left({\frac {x}{\lambda }}\right)^{k-1}e^{-(x/\lambda )^{k}}&x\geq 0\\0&x<0\end{cases}}$ \|	${\frac {\alpha \,x_{\mathrm {m} }^{\alpha }}{x^{\alpha +1}}}{\text{ for }}x\geq x_{m}$	${\frac {1}{2^{\frac {k}{2}}\Gamma \left({\frac {k}{2}}\right)}}\;x^{{\frac {k}{2}}-1}e^{-{\frac {x}{2}}}\,$
cdf	${\tfrac {1}{2}}\left[1+\operatorname {erf} \left({\frac {x-\mu }{\sigma {\sqrt {2}}}}\right)\right]$	${\frac {1}{2}}+{\frac {1}{2}}\operatorname {erf} {\Big [}{\frac {\ln x-\mu }{{\sqrt {2}}\sigma }}{\Big ]}$	${\begin{matrix}{\frac {1}{2}}+x\Gamma \left({\frac {\nu +1}{2}}\right)\times \\[0.5em]{\frac {\,_{2}F_{1}\left({\frac {1}{2}},{\frac {\nu +1}{2}};{\frac {3}{2}};-{\frac {x^{2}}{\nu }}\right)}{{\sqrt {\pi \nu }}\,\Gamma \left({\frac {\nu }{2}}\right)}}\end{matrix}}$ where ₂F₁ is the hypergeometric function	${\begin{cases}{\frac {1}{2}}\exp \left({\frac {x-\mu }{b}}\right)&{\mbox{if }}x<\mu \\[8pt]1-{\frac {1}{2}}\exp \left(-{\frac {x-\mu }{b}}\right)&{\mbox{if }}x\geq \mu \end{cases}}$	${\begin{cases}1-e^{-(x/\lambda )^{k}}&x\geq 0\\0&x<0\end{cases}}$	$1-\left({\frac {x_{\mathrm {m} }}{x}}\right)^{\alpha }{\text{ for }}x\geq x_{m}$	${\frac {1}{\Gamma \left({\frac {k}{2}}\right)}}\;\gamma \left({\frac {k}{2}},\,{\frac {x}{2}}\right)$
mean	$\quad \mu$	$\exp(\mu +\sigma ^{2}/2)$	0 for $\nu >1$ , otherwise undefined	$\mu$	$\lambda \,\Gamma (1+1/k)\,$	${\begin{cases}\infty &{\text{for }}\alpha \leq 1\\{\frac {\alpha \,x_{\mathrm {m} }}{\alpha -1}}&{\text{for }}\alpha >1\end{cases}}$	$k$
variance	$\quad \sigma ^{2}$	$[\exp(\sigma ^{2})-1]\exp(2\mu +\sigma ^{2})$	$\textstyle {\frac {\nu }{\nu -2}}$ for $\nu >2$ , ∞ for $1<\nu \leq 2$ , otherwise undefined	$2b^{2}$	$\lambda ^{2}\left[\Gamma \left(1+{\frac {2}{k}}\right)-\left(\Gamma \left(1+{\frac {1}{k}}\right)\right)^{2}\right]\,$	${\begin{cases}\infty &{\text{for }}\alpha \in (0,2]\\{\frac {x_{\mathrm {m} }^{2}\alpha }{(\alpha -1)^{2}(\alpha -2)}}&{\text{for }}\alpha >2\end{cases}}$	$2k$
skewness	$\quad 0$	$(e^{\sigma ^{2}}\!\!+2){\sqrt {e^{\sigma ^{2}}\!\!-1}}$	0 for $\nu >3$ , otherwise undefined	$0$	${\frac {\Gamma (1+3/k)\lambda ^{3}-3\mu \sigma ^{2}-\mu ^{3}}{\sigma ^{3}}}$	${\frac {2(1+\alpha )}{\alpha -3}}\,{\sqrt {\frac {\alpha -2}{\alpha }}}{\text{ for }}\alpha >3$	$\scriptstyle {\sqrt {8/k}}\,$
support	$x\in \mathbb {R}$	$x\in (0,+\infty )$	x ∈ (−∞; +∞)	$x\in (-\infty ;+\infty )\,$	$x\in [0,+\infty )\,$	$x\in [x_{\mathrm {m} },+\infty )$	$x\in [0,+\infty )$
median	$\quad \mu$	$\exp(\mu )$	0	$\mu$	$\lambda (\ln 2)^{1/k}\,$	$x_{\mathrm {m} }{\sqrt[{\alpha }]{2}}$	$\approx k{\bigg (}1-{\frac {2}{9k}}{\bigg )}^{3}$
quantile	$\mu +\sigma {\sqrt {2}}\operatorname {erf} ^{-1}(2F-1)$	-	-	-	-	-	-