Did you know that Helmert’s distribution is just another name for the chi-square distribution? It’s named after F.R. Helmert, who proved the general reproductive property of chi-square distributions.
Helmert’s most noted contribution was establishing that if a set of independent, normally distributed random variables X1, X2, …, Xn, then the chi-square variable is distributed as a chi-square variable with n – 1 degrees of freedom. This chi-square variable is statistically independent of “X-bar” (the sample mean). Helmert also proved that the sample mean and variance are independent in 1785. Helmert’s transformation allows us to obtain a set of k – 1 new independent and identically distributed normally distributed samples with μ = 0 and the same variance of the original distribution.
So what does all this mean? Essentially, Helmert’s discovery allows us to more effectively study normally distributed variables. His findings are still used today in many different fields, including statistics and finance. The next time you’re crunching numbers, think of Helmert and his impact on the world of data analysis!
Helmert’s Distribution: Conclusion
F.R. Helmert was a German mathematician and statistician who made significant contributions to the fields of statistics and probability theory. His most notable discovery was establishing that if a set of independent, normally distributed random variables X1, X2, …, Xn, then the chi-square variable is distributed as a chi-square variable with n – 1 degrees of freedom. Thanks to Helmert’s groundbreaking work, we are now able to more effectively study normally distributed variables.
References
Helmert, F. R. (1876). Die Genauigkeit der Formel von Peters zue Berechnung des wahrscheinlichen Beobachtungsfehlers directer Beobachtungen gleicher Genauigkeit, Astronomische Nachrichten, 88, columns 113-120;
Helmert, F. R. (1875). ~ber die Berechnung der wahrscheinlichen Fehlers aus einer endlichen Anzahl wahrer Beobachtungsfehler, Zeitschrift fur angewandte Mathe[1]matik und Physik, 20, 300-303.
Kruskal, W. Helmert’s Distribution. The American Mathematical Monthly. Vol. 53, No. 8 (Oct., 1946), pp. 435-438 (4 pages). Published By: Taylor & Francis, Ltd
The math genealogy project: F Helman.
Shakti, P. (2022). About Helmert’s Distribution.
Wu, M. et al. (2019). Differentiable Antithetic Sampling for Variance Reduction in Stochastic Variational Inference.