Distribution

$X\left(a,b\right)$

Mean

${\mu }_X=5$

Variance

${\sigma }_X^2=1$

Probability ${\Delta }_X$ $P\left(\left|x-{\mu }_X\right|\le {\sigma }_X\right)$

Normal $N\left({\mu }_X,{\sigma }_X^2\right)$

${\mu }_X$

${\sigma }_X^2$

$\text{erf}\left(2^{-\frac{1}{2}}\right)\simeq 0.6827$

Log-Normal $\Lambda \left(m,s^2\right)$

$\begin{array}{l}m=1.5898\\ s=0.1980\end{array}$

$\exp \left(m+\frac{1}{2}{s}^2\right)$

${\text{e}}^{2m}\left[{\text{e}}^{2s^2}-{\text{e}}^{s^2}\right]$

0.6940

Uniform $U\left(a,b\right)$

$\begin{array}{l}a=3.2679\\ b=6.7321\end{array}$

$\frac{1}{2}\left(a+b\right)$

$\frac{1}{12}{\left(b-a\right)}^2$

$\frac{1}{\sqrt{3}}\simeq 0.5774$

Laplace $La\left(\mu ,\beta \right)$

$\begin{array}{l}\mu =5\\ \beta =2^{-\frac{1}{2}}\simeq 0.7071\end{array}$

$\mu$

$2{\beta }^2$

$1-{\text{e}}^{-\sqrt{2}}\simeq 0.7569$