Gaussians

Gaussians refer to the Gaussian family in mathematics and statistics, named after Carl Friedrich Gauss. They describe smooth, bell-shaped curves that arise in many contexts due to symmetry, central limit phenomena, and convenient algebraic properties.

In one dimension, the Gaussian function is g(x) = A exp(- (x - μ)^2 / (2σ^2)). It is symmetric about μ and has width determined by σ > 0. Choosing A = 1/(σ√(2π)) yields the probability density function of the normal distribution with mean μ and standard deviation σ. The standard normal distribution, with μ = 0 and σ = 1, has density φ(x) = (1/√(2π)) exp(-x^2/2).

In n dimensions, the Gaussian density with mean vector μ and covariance matrix Σ is f(x) = (1/((2π)^{n/2} |Σ|^{1/2})) exp(-1/2 (x-μ)^T Σ^{-1} (x-μ)). Gaussian models with affine means preserve Gaussian form, and the sum of independent Gaussian variables is Gaussian. The product of two Gaussians is proportional to a Gaussian, and the Fourier transform of a Gaussian is again a Gaussian. The integral ∫ exp(-x^2) dx over the real line equals √π.

Applications span statistics, physics, and engineering. Gaussians model measurement errors, appear in the central limit theorem, and are used in signal and image processing (Gaussian blur), quantum mechanics (Gaussian wave packets), and many theoretical analyses due to their tractable algebraic properties.