  \, e^{-\frac{(x - \mu)^2}{2 \sigma^2}} | cdf = \frac12\left[1 + \operatorname{erf}\left( \frac{x-\mu}{\sigma\sqrt{2}}\right)\right] | quantile = \mu+\sigma\sqrt{2}\,\operatorname{erf}^{-1}(2f-1) | mean =  | median =  | mode =  | variance = \sigma^2\, | skewness = 0 | kurtosis = 0 | entropy = \frac12 \ln(2 \pi e \, \sigma^2) | mgf = \exp\{ \mu t + \frac{1}{2}\sigma^2t^2 \} | char = \exp \{ i\mu t - \frac{1}{2}\sigma^2 t^2 \} | fisher = \begin{pmatrix}1/\sigma^2&0\\0&1/(2\sigma^4)\end{pmatrix} | conjugate prior = normal distribution }} in probability theory, the normal (or gaussian) distribution is a very commonly occurring continuous probability distributionâa function that tells the probability that any real observation will fall between any two real limits or real numbers, as the curve approaches zero on either side. normal distributions are extremely important in statistics and are often used in the natural and social sciences for real-valued random variables whose distributions are not known.normal distribution, gale encyclopedia of psychology the normal distribution is immensely useful because of the central limit theorem, which states that, under mild conditions, the mean of many random variables independently drawn from the same distribution is distributed approximately normally, irrespective of the form of the original distribution: physical quantities that are expected to be the sum of many independent processes (such as measurement errors) often have a distribution very close to the normal.lyon, a. (2014). why are normal distributions normal?, the british journal for the philosophy of science. moreover, many results and methods (such as propagation of uncertainty and least squares parameter fitting) can be derived analytically in explicit form when the relevant variables are normally distributed. the gaussian distribution is sometimes informally called the bell curve. however, many other
Conjugate_prior_distributions Continuous_distributions Distributions_with_conjugate_priors Exponential_family_distributions Normal_distribution Probability_distributions Stable_distributions