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In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is [ 2 ] [ 3 ] f ( x ) = 1 2 π σ 2 e − ( x − μ ) 2 2 σ 2 . {\displaystyle f(x)={\frac {1}{\sqrt {2\pi \sigma ^{2 ...
Gaussian functions are often used to represent the probability density function of a normally distributed random variable with expected value μ = b and variance σ 2 = c 2. In this case, the Gaussian is of the form [1]
The normal distribution, also called the Gaussian or the bell curve. It is ubiquitous in nature and statistics due to the central limit theorem : every variable that can be modelled as a sum of many small independent, identically distributed variables with finite mean and variance is approximately normal.
A different technique, which goes back to Laplace (1812), [3] is the following. Let = =. Since the limits on s as y → ±∞ depend on the sign of x, it simplifies the calculation to use the fact that e −x 2 is an even function, and, therefore, the integral over all real numbers is just twice the integral from zero to infinity.
[2] In the social sciences , a result may be considered statistically significant if its confidence level is of the order of a two-sigma effect (95%), while in particle physics and astrophysics , there is a convention of requiring statistical significance of a five-sigma effect (99.99994% confidence) to qualify as a discovery .
In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional normal distribution to higher dimensions.
A Gaussian minus exponential distribution has been suggested for modelling option prices. [20] If such a random variable Y has parameters μ , σ , λ , then its negative -Y has an exponentially modified Gaussian distribution with parameters -μ , σ , λ , and thus Y has mean μ − 1 λ {\displaystyle \mu -{\tfrac {1}{\lambda }}} and variance ...
In the previous two integrals, n!! is the double factorial: for even n it is equal to the product of all even numbers from 2 to n, and for odd n it is the product of all odd numbers from 1 to n; additionally it is assumed that 0!! = (−1)!! = 1.