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In probability theory, an exponentially modified Gaussian distribution (EMG, also known as exGaussian distribution) describes the sum of independent normal and exponential random variables. An exGaussian random variable Z may be expressed as Z = X + Y , where X and Y are independent, X is Gaussian with mean μ and variance σ 2 , and Y is ...
An application for Peirce's criterion is removing poor data points from observation pairs in order to perform a regression between the two observations (e.g., a linear regression).
Examples of the Crystal Ball function. The Crystal Ball function, named after the Crystal Ball Collaboration (hence the capitalized initial letters), is a probability density function commonly used to model various lossy processes in high-energy physics. It consists of a Gaussian core portion and a power-law low-end
The generalized normal log-likelihood function has infinitely many continuous derivates (i.e. it belongs to the class C ∞ of smooth functions) only if is a positive, even integer. Otherwise, the function has ⌊ β ⌋ {\displaystyle \textstyle \lfloor \beta \rfloor } continuous derivatives.
The pseudo-Voigt profile (or pseudo-Voigt function) is an approximation of the Voigt profile V(x) using a linear combination of a Gaussian curve G(x) and a Lorentzian curve L(x) instead of their convolution. The pseudo-Voigt function is often used for calculations of experimental spectral line shapes.
"Table of zeros and Gaussian Weights of certain Associated Laguerre Polynomials and the related Hermite Polynomials". Mathematics of Computation. 18 (88): 598–616. doi: 10.1090/S0025-5718-1964-0166397-1. JSTOR 2002946. MR 0166397. Ehrich, S. (2002). "On stratified extensions of Gauss-Laguerre and Gauss-Hermite quadrature formulas".
Mathematically, the derivatives of the Gaussian function can be represented using Hermite functions. For unit variance, the n-th derivative of the Gaussian is the Gaussian function itself multiplied by the n-th Hermite polynomial, up to scale. Consequently, Gaussian functions are also associated with the vacuum state in quantum field theory.
SciPy (pronounced / ˈ s aɪ p aɪ / "sigh pie" [2]) is a free and open-source Python library used for scientific computing and technical computing. [3]SciPy contains modules for optimization, linear algebra, integration, interpolation, special functions, FFT, signal and image processing, ODE solvers and other tasks common in science and engineering.