Jaroszewicz, Szymon; Korzeń, Marcin Arithmetic operations on independent random variables: a numerical approach. (English) Zbl 1260.65010 SIAM J. Sci. Comput. 34, No. 3, A1241-A1265 (2012). Summary: Dealing with imprecise quantities is an important problem in scientific computation. Model parameters are known only approximately, typically in the form of probability density functions. Unfortunately there are currently no methods of taking uncertain parameters into account which would at the same time be easy to apply and highly accurate. An important special case is operations on independent random variables which occur frequently in obtaining confidence intervals for physical measurements and statistical estimators.In this paper we investigate the possibility of implementing arithmetic operations on independent random variables numerically. Equivalently, the problem can be viewed as propagating approximated probability density functions through arithmetic operations. We introduce a very broad family of distributions which is closed under the four arithmetic operations and taking powers. Furthermore we show that the densities of the distributions in the family can be effectively approximated using Chebyshev polynomials. A practical implementation is also provided, demonstrating the feasibility and usefulness of the approach.Several examples show applications in physical measurements, statistics, and probability theory, demonstrating very high numerical accuracy. These include an interesting problem related to combining independent measurements of a physical quantity, distributions of sample statistics of the Hill’s estimator for tail exponents, generalized \(\chi^2\) distribution, and others. The results are usually extremely accurate, in some cases more accurate than specialized solutions available in statistical packages, and can be achieved with very little effort. Cited in 3 Documents MSC: 65C60 Computational problems in statistics (MSC2010) 62F25 Parametric tolerance and confidence regions 62F10 Point estimation Keywords:uncertainty propagation; numerical examples; confidence intervals; physical measurements; statistical estimators; arithmetic operations; on independent random variables; distributions of sample statistics; Hill’s estimator; generalized \(\chi^2\) distribution Software:PaCAL; Chebfun; APPL; Matlab PDFBibTeX XMLCite \textit{S. Jaroszewicz} and \textit{M. Korzeń}, SIAM J. Sci. Comput. 34, No. 3, A1241--A1265 (2012; Zbl 1260.65010) Full Text: DOI