A method for fitting probability distributions to engineering properties of rock masses using Legendre orthogonal polynomials

A method is presented for fitting probability distributions of random variables describing the engineering properties of rock in this paper. The Legendre orthogonal polynomial (LPA method) with sample origin moments was used to approximate the PDF (probability density function) or CDF (cumulative distribution function) of rock random variable. It is distribution-free because no classical theoretical distributions are assumed in advance and the inference result provides a universal form of PDF. Six most commonly-used theoretical distributions were used to verify LPA method. Twenty groups’ random samples of traditional distribution by Monte Carlo method and four examples from observed data of mechanic parameters of rock were presented for illustrative purposes. The largest departure Dn of Kolmogorov–Smirnov test and the RE (percentage relative error) of estimation by LPA method is smaller than that of PDF fitted by standard methods of statistical inference. The results show the PDF of rock variables can be accurately derived from LPA method with sample moments and the proposed method is suitable for approximating probability distributions of rock random variable.