Gaussian concentration

For $X \sim \mathcal{N}(\mu,\sigma^2)$ : $\mathrm{Pr}[X=\mu \pm x] \sim \frac{1}{\sigma\sqrt{2\pi}}e^{-x^2/2\sigma^2}$

Gaussian function

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

Probability density function of normally distributed random variable, expected value $\mu = b$ , variance $\sigma^2 = c^2$ , $g(x)={\frac {1}{\sigma {\sqrt {2\pi }}}}\exp \left(-{\frac {1}{2}}{\frac {(x-\mu )^{2}}{\sigma ^{2}}}\right)$