On the Distribution of the Weighted Sum of Chi-Squared Variables

2021/09/14 by Ayse Unsal, Unsal, Ayse, Raymond Knopp +1
Decision Sciences · #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT) #Probabilistic and Robust Engineering Design #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.2203.11940

openalex publication_date 2021/09/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This paper presents the probability distribution function (p.d.f.) and cumulative distribution function (c.d.f.) of the weighted sum of central independent chi-squared random variables with non-zero weighs based on a method using moment generating functions. To obtain the p.d.f. and c.d.f. of such a function, we first derive the moment generating function of this weighted sum using the partial fractions decomposition or the residue method. The results cover the sum of two and three weighted chi-squared random variables, which can easily be adapted to more general cases.

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