Probability analysis method using Fast Fourier transform and its application

This paper presents an efficient probability analysis method using Fast Fourier transform taking into account that (1) a probability density function and its characteristic function are a pair of Fourier transforms, and (2) the characteristic function of the sum of statistically independent random variables is given by the product of the characteristic functions of each random variable. The efficient probability analysis method is extended for the sum of correlated non-normal random variables with correlation coefficients expressed by the power of a common value. Expressing the correlated basic variables in terms of two sets of uncorrelated random variables, the characteristic function of their sum can be estimated approximately by the characteristic functions of the correlated basic variables. The probability density function of the sum is obtained using Fast Fourier transform. The accuracy and applicability of the approximation method are investigated based on the estimation error of the cumulant and using numerical examples. Applications of the proposed method to various aspects of structural reliability analysis and design are illustrated.