gaussiana

Gaussiana, in mathematical terminology often referred to as the Gaussian or normal distribution, is a continuous probability distribution characterized by its bell-shaped curve. It is symmetric around its mean and is derived from or approximates a wide range of natural phenomena and measurement errors. The concept is named after Carl Friedrich Gauss, who contributed to its mathematical development and its use in error analysis.

In one dimension, the probability density function is f(x | μ, σ) = (1 / (σ sqrt(2π))) exp(- (x − μ)² / (2σ²)), where μ is the mean and σ is the standard deviation. The standard normal distribution arises when μ = 0 and σ = 1. In higher dimensions, the multivariate normal distribution generalizes this form with a mean vector μ and a covariance matrix Σ, leading to a density proportional to exp(-1/2 (x − μ)ᵀ Σ⁻¹ (x − μ)) and involving determinants of Σ.

Key properties include symmetry, unimodality, finite variance, and tails that decay exponentially. The central limit theorem explains why sums of many independent random effects tend to be approximately Gaussian, making the normal distribution a common model for measurement errors and natural variability. Estimation of its parameters is typically done via maximum likelihood or moment methods, with standard scores (z-scores) used to standardize observations.

Applications span statistics, natural and social sciences, engineering, and data analysis. Gaussian approximations underpin many statistical tests and models, and Gaussian filters are used for smoothing in image and signal processing. While widely applicable, real-world data may deviate from normality, necessitating transformation or robust methods.