Author(s)

Thomas M. Semkow^1,2, Nicole Freeman³, Umme-Farzana Syed¹, Douglas K. Haines¹, Abdul Bari¹, Abdul J. Khan¹, Kimi Nishikawa¹, Adil Khan¹, Adam G. Burn^1,2, Xin Li^1,2, Liang T. Chu^1,2

¹Wadsworth Center, New York State Department of Health, Albany, NY, USA.
²Department of Environmental Health Sciences, University at Albany, State University of New York, Rensselaer, NY, USA.
³Averill Park Central School District, Averill Park, NY, USA.

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.

Mathematical Induction, Laplace Transform, Gamma Distribution, Chi-Square Test, Gross Alpha-Beta, Drinking Water

Semkow, T. , Freeman, N. , Syed, U. , Haines, D. , Bari, A. , Khan, A. , Nishikawa, K. , Khan, A. , Burn, A. , Li, X. and Chu, L. (2019) Chi-Square Distribution: New Derivations and Environmental Application. Journal of Applied Mathematics and Physics, 7, 1786-1799. doi: 10.4236/jamp.2019.78122.