# Normal distreebution

In probability theory, the normal (or Gaussian) distreebution is a very common conteenous probability distreebution.

Notation Probability density functionThe reid curve is the staundart normal distreibution Cumulative distribution function ${\displaystyle {\mathcal {N}}(\mu ,\,\sigma ^{2})}$ μ ∈ R — mean (location)σ2 > 0 — variance (squerred scale) x ∈ R ${\displaystyle {\frac {1}{\sqrt {2\sigma ^{2}\pi }}}\,e^{-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}}}$ ${\displaystyle {\frac {1}{2}}\left[1+\operatorname {erf} \left({\frac {x-\mu }{\sigma {\sqrt {2}}}}\right)\right]}$ ${\displaystyle \mu +\sigma {\sqrt {2}}\,\operatorname {erf} ^{-1}(2F-1)}$ μ μ μ ${\displaystyle \sigma ^{2}\,}$ 0 0 ${\displaystyle {\tfrac {1}{2}}\ln(2\sigma ^{2}\pi \,e\,)}$ ${\displaystyle \exp\{\mu t+{\frac {1}{2}}\sigma ^{2}t^{2}\}}$ ${\displaystyle \exp\{i\mu t-{\frac {1}{2}}\sigma ^{2}t^{2}\}}$ ${\displaystyle {\begin{pmatrix}1/\sigma ^{2}&0\\0&1/(2\sigma ^{4})\end{pmatrix}}}$