Numerical estimation of basic arithmetical operations on bounded random variables
Abstract
The objective of the study is estimating probability density functions of arithmetic operations on random variables. There are many methods that use specific parametrical and non-parametrical models in order to obtain accurate results. However, there are not many studies on speed of convergence and computation complexity of these methods.
This paper introduces a new method of estimation used to obtain results on bounded random variables. The method is based on a new publication provided by Jaroszewicz and Korzen (2012). The algorithm uses numerical analysis techniques such as numerical integration and curve interpolation. Author's method is compared to the well-known Monte Carlo method.
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Introduction
Estimation of probability density function (PDF) has been a topic of many publications [1,2,3]. If we have PDF of random variable and want to perform some operations on it, we can use direct approach and calculate these operations as it is shown in [4]. Sometimes it is very difficult or even impossible to obtain exact result as an elementary function [5]. That is why we usually use techniques such as Monte Carlo method in order to get approximate result. Authors created method that comes from direct numerical approximation of integrals. The method is suitable and time efficient when we want to obtain the result of basic arithmetical operations on random variables.
Let $X$ and $Y$ be independent random variables with known PDFs fX(x), fY(x). Let g(X, Y) be the function on random variables. Our formal task is to obtain approximate PDF of g. Basic arithmetical operations consist of addition, subtraction, multiplication, and division. Our method is focused only on finding PDFs of X+Y, X-Y, XY, X/Y.
Conclusion
Authors developed the method to perform basic arithmetical operations on random variables. In comparison to well-known and commonly applied KDE method, it is faster and more adequate in simulations. The method is in prototype phase and authors are willing to put further effort in order to develop it, what is to minimize faster error measures. The results of our study are discussed focusing on convergence rate observed from results of the simulation.
In further work, authors want to examine the best relationship between integration accuracy- dx and number of chosen pointsnx. The method should give better result when we know how to choose points x and we apply integration methods in a more effective way. Authors also want to compare KDE and numerical method with different kernel estimators for bigger amount of samples. Authors are also willing to develop method to estimate PDFs of more complex operations on random variables.