Methods for Calculating the Probability Distribution of Sums of Independent Random Variables
ARMY ARMAMENT MUNITIONS AND CHEMICAL COMMAND ROCK ISLAND IL SYSTEMS ANALYSIS OFFICE
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This report surveys numerical methods for obtaining the probability distribution of a sum of statistically independent random variables. Study objectives are to investigate the relative accuracy and computational effort for each of the following methods a evaluation of closed-form solutions for particular cases, b discrete numerical convolution of probability densities, c Normal probability approximation to the distribution of a sum, d numerical inversion of the Laplace transform of the convolution, e Erlang approximation for convolutions of a two-parameter Weibull distribution, f convolution of probability densities using the FFT algorithm for calculating finite Fourier transforms, and g Monte-Carlo simulation. Methods are sketched for deriving analytic expressions for the distribution of the sum of RVs of certain distributions. Each numerical method is described and illustrated using RVs from several distributional forms, such as uniform, exponential, gamma, and Weibull, as well as mixture models. In terms of run time and accuracy, some methods are particularly suited to certain distributional forms. If problem applications are quite special and if the time for program coding as well as running is a consideration, Monte-Carlo simulation may be the preferred method. All computer source programs are listed in annexes.
- Statistics and Probability