Upper Bound Error of Estimated Probability Density Function of the Product of Two Normal Random Variables

Rifyan Nasution, Gianto, Roberd Saragih and Khreshna Syuhada ()
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Rifyan Nasution: Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, Bandung 42132, Indonesia
Gianto: Academy of Metrology and Instrumentation, Ministry of Trade, Sumedang 45362, Indonesia
Roberd Saragih: Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, Bandung 42132, Indonesia
Khreshna Syuhada: Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, Bandung 42132, Indonesia

Mathematics, 2025, vol. 13, issue 19, 1-25

Abstract: The probability density function (PDF) of the product of two normal random variables remains an open discussion. Researchers have proposed many forms of PDFs. Among these, two notable PDFs are an analytical solution with infinite summation and an integral form with transformation. For practical computation, they must be estimated. The form with infinite summation must be truncated to a finite summation, and the form still in integration must be computed numerically. As a result of this estimation, an error occurs in the value of the estimation. This paper derives upper bounds for the estimation error resulting from truncation and numerical approximation in integral calculations. The upper bound error between the exact PDF and the truncated PDF is expressed as a geometric series using Bessel function inequality and Stirling’s approximation. The geometric formula allows the quantification of the total truncation error to be determined. For the PDF, which is still in integration form, the trapezoidal rule is used for numeric calculation. Hence, the error can be determined using the error-bound formula. The two estimated PDFs have their own advantages and disadvantages. The truncated PDF gives a relatively small upper bound value compared to the numerical calculation integral form PDF for a small value domain. However, the truncated PDF fails to perform for a large value domain, and only the integral form PDF can be used. The error for the estimation is applied to the conventional mass measurement. The results demonstrate that the error can be controlled through an analytical approach.

Keywords: Bessel function; Stirling’s formula; dependent random variables; trapezoidal rule; bisection method (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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