Gaussian Function

In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function of the base form:

for arbitrary real constants , , and non-zero . It is names after the mathematician Carl Friedrich Gauss.

The graph of a Gaussian is a characteristic symmetric bell curve shape. The parameter is the height of the curve's peak, is the position of the center of the peak, and (the standard deviation) 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 and variance . In this case, the gaussian is of the form:

Gaussian function uses:

• statistics to describe normal distributions
• signal processing to define Gaussian filters
• image processing where two-dimensional Gaussians are used for Gaussian blurs
• in mathematics to solve heat equations and diffusion equations and to define the Weierstrass transform

• The integral of a Gaussian function is 1 only if (the normalizing constant), and in this case the Gaussian is the probability density function of a normally distributed random variable with expected value and variance .