Gaußsche

Gaußsche is a German adjective derived from the name Gauss (Carl Friedrich Gauss) and is used in mathematics, statistics, and related fields to denote concepts associated with his work. In practice, the term most often refers to the Gaussian, or normal, distribution and to Gaussian functions, which model many naturally occurring phenomena and serve as standard tools in data analysis and error theory. The word appears in phrases such as gaußsche Verteilung or gaußsche Funktion.

The Gaussian distribution is a continuous probability distribution characterized by its bell-shaped curve, symmetric around its mean μ, and defined by the probability density function f(x) = (1/(σ√(2π))) exp(-(x-μ)^2/(2σ^2)), where σ > 0 is the standard deviation. Its standard form, the standard normal distribution, has μ = 0 and σ = 1. Key properties include symmetry, unimodality, and that many statistics (means, variances) are determined by μ and σ. The central limit theorem explains its frequent appearance as the limiting distribution of sums of independent random variables.

Applications and history: The Gaussian distribution is widely used to model measurement errors, natural and social phenomena, and signals in science and engineering. Gauss contributed to error theory and the method of least squares, underpinning many gaußsche approaches in data fitting. The term also extends to Gaussian functions in signal processing and differential equations, and to concepts such as Gaussian blur in image processing.