Numerical Methods for Approximating Continuous Probability Density Functions, over [0, ∞), Using Moments

Three numerical methods are presented for the reconstruction of a continuous probability density function f(x) from given values of the moments of the distribution. The first method is obtained by assuming that f(x) may be expanded as an infinite series of generalized Laguerre polynomials $Lna(x)$. The use of ordinary Laguerre polynomials, corresponding to the particular choice α = 0, is related to a second method involving the numerical inversion of a Laplace transform. In the third method the principle of maximization of entropy, subject to the known moment constraints, is used to reconstruct f(x). The type of fit to be expected from each method is illustrated by numerical examples.