Gaussiane

Gaussiane is a term found in mathematical Latin and some scholarly writings to refer to Gaussian objects, especially those named after Carl Friedrich Gauss. In modern English, the singular Gaussian and the plural Gaussians are more common, but Gaussiane may appear in historical texts or Latin translations to denote a family of Gaussian curves or distributions.

The central members of the Gaussiane are the Gaussian function and the related normal distribution used in probability theory. A Gaussian function on the real line can be written as f(x) = (1/(σ√(2π))) exp(-(x-μ)^2/(2σ^2)), where μ is the mean and σ is the standard deviation. The standard normal distribution has μ = 0 and σ = 1. The Gaussian family is characterized by its bell-shaped curve and its parameterization by location μ and scale σ.

Key properties include symmetry about μ and the fact that Gaussian objects are closed under certain operations. The sum (or linear combination) of independent Gaussian variables is again Gaussian, and the Fourier transform of a Gaussian is another Gaussian. The Gaussian integral ∫ exp(-x^2) dx = √π underpins many theoretical results, and convolution of Gaussians yields a Gaussian with variance equal to the sum of the variances.

Applications are widespread: statistics and data analysis use Gaussian models for measurement errors; in signal and image processing, Gaussian functions model noise and produce Gaussian blur; in physics and engineering, Gaussians describe a variety of natural and experimental phenomena.

Historical notes: Gauss introduced the normal distribution in the context of measurement errors, leading to its naming. The Latin plural Gaussiane appears in older texts to denote the family as a whole.