توزیع نرمال

${\displaystyle f(x;\,\mu ,\sigma ^{2})={\frac {1}{\sqrt {2\pi \sigma ^{2}}}}\,e^{-(x-\mu )^{2}\!/(2\sigma ^{2})},\qquad x\in \mathbb {R} .}$

• ${\displaystyle \mathrm {E} {\big [}(X-\mu )^{p}{\big ]}={\begin{cases}0&{\text{if }}p{\text{ is odd,}}\\\sigma ^{p}\,(p-1)!!&{\text{if }}p{\text{ is even.}}\end{cases}}}$

• ${\displaystyle \operatorname {E} {\big [}|X-\mu |^{p}{\big ]}=\sigma ^{p}(p-1)!!\cdot {\begin{cases}{\sqrt {2/\pi }}&{\text{if }}p{\text{ is odd}},\\1&{\text{if }}p{\text{ is even}}.\end{cases}}}$

${\displaystyle Z={\frac {X-\mu }{\sigma }}}$

${\displaystyle X\sim N(2,9)}$

${\displaystyle P(X<3)=?}$

${\displaystyle P(X<3)=P({\frac {X-\mu }{\sigma }}<{\frac {3-2}{3}})\approx P(Z<0.33)=0.6293}$