Gaussian distribution

Definition (Gaussian concentration)

For $X \sim \mathcal{N}(\mu,\sigma^2)$ :

\mathrm{Pr}[X=\mu \pm x] \sim \frac{1}{\sigma\sqrt{2\pi}}e^{-x^2/2\sigma^2}

See Gaussian tail bound

in terms of $G$ function,
#incomplete

Probability density function (pdf) of Gaussian distribution

The probability density function of a normally distributed random variable, with expected value $\mu = b$ , and variance $\sigma^2 = c^2$ , is

g(x)={\frac {1}{\sigma {\sqrt {2\pi }}}}\exp \left(-{\frac {1}{2}}{\frac {(x-\mu )^{2}}{\sigma ^{2}}}\right)

Gaussian function

Gaussian function takes the form $f(x) = \exp(-x^2)$ ,
and parametric extension $f(x)=a\exp \left(-{\frac {(x-b)^{2}}{2c^{2}}}\right)$ , $a,b,c \in \mathbb{R}$ .

Jointly Gaussian

two variables

$(X,Y) = \mathcal{N}(\eta_1, \eta_2, \sigma_1^2,\sigma_2^2,r)$
where $|r|\leq 1$ (correlation coefficient)

pdf given by
#incomplete

References: