vigafunktsioon

Vigafunktsioon, often translated as "vigafunction" or "error function" in English, is a special function that arises in probability and statistics, particularly in relation to the normal distribution. It is defined as a specific integral of the Gaussian function. The error function, denoted by erf(x), is given by the integral from negative infinity to x of the Gaussian function e^(-t^2) dt, scaled by a constant factor. A related function, the complementary error function, erfc(x), is defined as 1 - erf(x).

The primary utility of the vigafunktsioon lies in its connection to the cumulative distribution function (CDF) of the standard normal distribution. The CDF of the standard normal distribution, often denoted by Φ(x), can be expressed in terms of the error function. Specifically, Φ(x) = 0.5 * [1 + erf(x / sqrt(2))]. This relationship allows for the calculation of probabilities associated with a normal distribution using values of the error function.

The error function is an odd function, meaning erf(-x) = -erf(x), and its value ranges from -1 to 1. It is frequently encountered in fields such as physics, engineering, and signal processing, where phenomena are modeled using the normal distribution. While it does not have a simple closed-form expression in terms of elementary functions, its values can be approximated or found in tables and computational software. The vigafunktsioon plays a crucial role in quantifying cumulative probabilities and understanding the behavior of random variables that follow a Gaussian distribution.