Normal distribution

probability density function

cumulative probability function

The normal distribution is a key distribution in the field of probability. It is also known as the Gaussian distribution, after mathematician Carl Gauss, and the bell curve. The normal probability density function (PDF) is

$f(x)={\frac {1}{\sqrt {2\pi \sigma ^{2}}}}\exp \left[-{\frac {1}{2\sigma ^{2}}}\left(x-\mu \right)^{2}\right]$

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu} is the mean and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma^2} is the variance.

If $\mu =0$ and $\sigma =1$, the distribution is called the standard normal distribution, often denoted by $\phi$:

$\phi (x)={\frac {1}{\sqrt {2\pi }}}\exp(-{\frac {1}{2}}x^{2}).$

See also

• Central limit theorem