Method for Deriving Probability Distributions with Gamma Functions

摘要

Based on an exponential integral formula recently derived by the authors, asubstitution method is presented for determining a probability density function and moments for a useful class of (0, infinity) exponentials frequently encountered in signal processing. The generalization of the method is elaborated to account for models of (-infinity, infinity) exponentials. The generalized density function and moments relation are given as integral formulas specifically calculated from their definitions in probability theory. Several examples of the method are provided, including an example for modeling random signals and deterministic acoustic signals in underwater acoustics. The generalization of the method for multivariate distributions is presented.