What is the Gaussian function?

The Gaussian function, also known as the Gaussian distribution or the normal distribution, is a mathematical function that describes the shape of a symmetrical bell curve. It is named after the mathematician Carl Friedrich Gauss, who extensively studied its properties.

The Gaussian function is defined by its probability density function (PDF), which is given by the equation:

f(x) = (1 / σ√(2π)) * e^(-(x - μ)² / (2σ²))

In this equation:

x represents the variable of interest.
μ (mu) is the mean of the distribution, which determines the center of the curve.
σ (sigma) is the standard deviation, which controls the spread or width of the curve.
e is the base of the natural logarithm, approximately equal to 2.71828.
The Gaussian function is symmetric around its mean μ and has a peak value at x = μ. The standard deviation σ determines the width of the curve. A smaller standard deviation produces a narrower curve, while a larger standard deviation leads to a wider curve.

The Gaussian distribution is widely used in statistics, probability theory, and various fields of science and engineering. It is often employed to model real-world phenomena that exhibit randomness and are influenced by multiple factors. The central limit theorem also states that under certain conditions, the sum or average of a large number of independent and identically distributed random variables will have an approximately Gaussian distribution.