ERROR FUNCTION Meaning and
Definition
-
The error function, denoted as erf(x), is a mathematical function commonly used in mathematics, statistics, and engineering. It is defined as the integral of the Gaussian or normal distribution from zero to a given value (x) multiplied by two divided by the square root of pi (√π). The error function is often considered the cumulative distribution function (CDF) of a standard normal distribution.
The error function is primarily used to calculate the probability of a variable within a specified range. It describes the probability that a normally distributed random variable will take on a value between negative infinity and x. In other words, it quantifies the error between a given value and the nearest integer.
The error function has applications in various fields such as signal processing, physics, and probability theory. It is particularly useful in statistics and data analysis as it helps in modeling and characterizing random errors or uncertainties in measurements or observed data.
The error function is a continuous and differentiable function that ranges from -1 to 1. It is an odd function, symmetric about the origin, and approaches -1 and 1 asymptotically as x tends to negative and positive infinity, respectively.
The error function can be calculated using various numerical methods, such as Taylor series expansion or approximation techniques, and it is implemented in various mathematical software libraries for easy computation.
Etymology of ERROR FUNCTION
The term "error function" is derived from the mathematical function called the "Gauss error function" or simply the "error function" (denoted as erf(x)). The term "error" refers to the deviation or discrepancy between an observed or measured value and its expected or true value. In the context of statistics, the error function is commonly used to quantify the probability of an event occurring within a certain range. It is named after the German mathematician Friedrich Bessel who introduced the function in 1817.
