: Matemáticas: Gaussian function

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Gaussian function

In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function of the base form ${\\displaystyle f(x)=\\exp(-x^{2})}$ and with parametric extension ${\\displaystyle f(x)=a\\exp \\left(-{\ rac {(x-b)^{2}}{2c^{2}}}\ ight)}$ for arbitrary real constants $a$ , $b$ and non-zero $c$ . It is named after the mathematician Carl Friedrich Gauss. The graph of a Gaussian is a characteristic symmetric "bell curve" shape. The parameter $a$ is the height of the curve's peak, $b$ is the position of the center of the peak, and $c$ (the standard deviation, sometimes called the Gaussian RMS width) controls the width of the "bell". Gaussian functions are often used to represent the probability density function of a normally distributed random variable with expected value $μ = b$ and variance $σ 2 = c 2$ . In this case, the Gaussian is of the form ${\\displaystyle g(x)={\ rac {1}{\\sigma {\\sqrt {2\\pi }}}}\\exp \\left(-{\ rac {1}{2}}{\ rac {(x-\\mu )^{2}}{\\sigma ^{2}}}\ ight).}$ Gaussian functions are widely used in statistics to describe the normal distributions, in signal processing to define Gaussian filters, in image processing where two-dimensional Gaussians are used for Gaussian blurs, and in mathematics to solve heat equations and diffusion equations and to define the Weierstrass transform.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Gaussian_function)