Extensão intervalar para as variáveis aleatórias com distribuições uniforme, normal, gama, exponencial e pareto

Abstract

When working with floating point numbers the result is only an approximation of a real value and errors generated by rounding or by instability of the algorithms can lead to incorrect results. We can’t affirm the accuracy of the estimated answer without the contribution of an error analysis. Using intervals for the representation of real numbers, it is possible to control this error propagation, because intervals results carry with them the security of their quality. To obtain the numerical value of the probability density functions of continuous random variables with distributions Uniform, Exponential, Normal, Gamma and Pareto is necessary to use numerical integration, once the primitive of the integral do not always is simple to obtain. Moreover, the result is obtained by approximation and therefore affected by truncation or rounding errors. In this context, this paper has three main objectives: i) define probability density function of variables with Gamma and Pareto distributions in the interval form completing the already existing definitions in the literature for the Uniform, Exponential and Normal distributions; ii) analyze the quality of wrappers intervals for random variables with distributions Uniform, Exponential, Normal, Gamma and Pareto, through results generated from algorithms implemented in the Python language, using the interval package IntPy; iii) analyze the computational complexity of computing the probability density functions with Uniform, Exponential, Normal, Gamma and Pareto distributions in the real and interval forms. Through these goals, make sure that by using interval arithmetic to calculate the probability density function of the random variables with distributions, it is possible to have an automatic error control with reliables boundaries, and, at least, keep the existing computational effort in the calculation using the real arithmetic.