Chi-Square Distribution: New Derivations and Environmental Application

• 作者: Abdul Bari Abdul J. Khan Adam G. Burn Adil Khan Douglas K. Haines Kimi Nishikawa Liang T. Chu Nicole Freeman Thomas M. Semkow Umme-Farzana Syed Xin Li
• 来源期刊: 应用数学与应用物理(英文)
• 学科: 医学
• 关键词: Mathematical Induction LAPLACE Transform GAMMA Distribution CHI-SQUARE Test GROSS Alpha-Beta DRINKING Water
• 年，卷（期）: 2019,（8）
• 页码范围: 1786-1799
• 页数: 14页
• 分类号: R73

摘要

We describe two new derivations of the chi-square distribution. The first derivation uses the induction method, which requires only a single integral to calculate. The second derivation uses the Laplace transform and requires minimum assumptions. The new derivations are compared with the established derivations, such as by convolution, moment generating function, and Bayesian inference. The chi-square testing has seen many applications to physics and other fields. We describe a unique version of the chi-square test where both the variance and location are tested, which is then applied to environmental data. The chi-square test is used to make a judgment whether a laboratory method is capable of detection of gross alpha and beta radioactivity in drinking water for regulatory monitoring to protect health of population. A case of a failure of the chi-square test and its amelioration are described. The chi-square test is compared to and supplemented by the t-test.