Normal Distribution (N₍μ,σ²₎) ○◂｜_append｜___｜20260628182425-01-•

Normal distribution	…	…	…
Probability density functionThe red curve is the standard normal distribution.	…	…	…
Cumulative distribution function	…	…	…
Notation	$\mathcal{N}(\mu ,{\sigma }^2)$	…	…
Parameters	$\mu \in \mathbb{R}$ = mean (location) ${\sigma }^2\in {\mathbb{R}}_{>0}$ = variance (squared scale)	…	…
Support	$x\in \mathbb{R}$	…	…
PDF	$\frac{1}{\sqrt{2\pi {\sigma }^2}}{e}^{-\frac{(x-\mu {)}^2}{2{\sigma }^2}}$	…	…
CDF	$\Phi \left(\frac{x-\mu }{\sigma }\right)=\frac{1}{2}\left[1+\mathrm{erf}\left(\frac{x-\mu }{\sigma \sqrt{2}}\right)\right]$	…	…
Quantile	$\mu +\sigma \sqrt{2}{\mathrm{erf}}^{-1}(2p-1)$	…	…
Mean	$\mu$	…	…
Median	$\mu$	…	…
Mode	$\mu$	…	…
Variance	${\sigma }^2$	…	…
MAD	$\sigma \sqrt{2}{\mathrm{erf}}^{-1}(1/2)$	…	…
AAD	$\sigma \sqrt{2/\pi }$	…	…
Skewness	$0$	…	…
Excess kurtosis	$0$	…	…
Entropy	$\frac{1}{2}\log (2\pi e{\sigma }^2)$	…	…
MGF	$\exp (\mu t+{\sigma }^2{t}^2/2)$	…	…
CF	$\exp (i\mu t-{\sigma }^2{t}^2/2)$	…	…
Fisher information	$\mathcal{I}(\mu ,\sigma )=\left(\begin{array}{cc}1/{\sigma }^2& 0\\ 0& 2/{\sigma }^2\end{array}\right)$ $\mathcal{I}(\mu ,{\sigma }^2)=\left(\begin{array}{cc}1/{\sigma }^2& 0\\ 0& 1/(2{\sigma }^4)\end{array}\right)$	…	…
Kullback–Leibler divergence	$\frac{1}{2}\left\{{\left(\frac{{\sigma }_0}{{\sigma }_1}\right)}^2+\frac{({\mu }_1-{\mu }_0{)}^2}{{\sigma }_1^2}-1+\ln \frac{{\sigma }_1^2}{{\sigma }_0^2}\right\}$	…	…

(echo:: @ ᯤ)