Gaussfüggvényeket

Gaussfüggvényeket, commonly referred to as Gaussian functions or normal distributions, are a family of continuous probability distributions that are fundamental in statistics and many other fields. The characteristic bell shape of the Gaussian function makes it easily recognizable. Its mathematical form is defined by two parameters: the mean ($\mu$) and the standard deviation ($\sigma$). The mean determines the center of the distribution, indicating where the peak of the bell curve lies, while the standard deviation controls the spread or width of the curve. A smaller standard deviation results in a narrower, taller peak, while a larger standard deviation leads to a wider, flatter curve.

The probability density function (PDF) of a Gaussian function is given by the formula:

$f(x | \mu, \sigma^2) = \frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}$

Gaussian functions are ubiquitous due to the Central Limit Theorem, which states that the sum (or average) of a large number of independent and identically distributed random variables will tend to be normally distributed, regardless of the original distribution of those variables. This property makes them incredibly useful for modeling various natural phenomena, such as heights of people, measurement errors, and biological data. They are also extensively used in signal processing, image filtering (e.g., Gaussian blur), and machine learning algorithms, particularly in areas like regression and classification. The standardized form, where the mean is 0 and the standard deviation is 1, is known as the standard normal distribution.