The error function and normal distribution
The error function erf(x) is defined as the integral of the Gaussian function from 0 to x, scaled so that erf(∞) = 1. It appears throughout probability, statistics, and physics wherever normal distributions or diffusion processes arise.
The standard normal CDF Φ(x) is directly related to erf: Φ(x) = 0.5 × (1 + erf(x/√2)). This connection makes erf essential for computing probabilities from z-scores.
erf(x) = (2/√π) ∫₀ˣ e^(−t²) dt
Definition of the error function as a definite integral.
Φ(x) = ½[1 + erf(x/√2)]
Standard normal CDF in terms of the error function.