Probability Density Functions

Abstract

Probability density functions are defined, and the main statistical indicators (average, variance, etc.) are extended from the discrete to the continuous case. The problem of determining the probability distribution under variable transformation is presented, and the analogies and differences with the discrete case are discussed. The main continuous probability distribution functions are presented, with their properties: uniform, Gaussian, chi-square, log normal, gamma and beta distribution. Some distributions specifically of interest in physics are also presented: Breit–Wigner, Argus Crystal Ball, Landau. The central limit theorem is presented. Probability density functions in multiple dimensions are discussed, with emphasis on the difference between independent and uncorrelated variables. The particular case of a Gaussian distribution in multiple dimensions, with possible correlation terms, is presented.