# Helmert’s Distribution

Helmert’s distribution is another name for the chi-square distribution. It is 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

1. Is distributed as a chi-square variable with n – 1 degrees of freedom
2. 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. Given a set of xk normally distributed i.i.d. variables, the transformation to yj random variables can be defined as 

Because of this important contribution to sampling theory, Kruskal  recommended naming the joint distribution of the two random variables “Helmert’s Distribution”. However, the recommendation did not take, and we still refer to Helmert’s result as the chi-square.

Helmert’s distribution is important in sampling theory because it leads to 

1. Statistical independence of sample mean (μ) and sample standard deviation (σ).
2. Separate single distributions of μ and σ.
3. Distribution of Student’s t.

Helmert obtained his distribution by setting up a pair of linear transformations. These transformed the joint distribution of individual observation errors into a joint distribution of errors for μ and σ; any dummy variables could be integrated out.

## 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. (1876). ~ber die Wahrscheinlichkeit der Potenzsummen der Beobachtungsfehler und iiber einige damit in Zusammenhange stehende Fragen, Zeitschrift fir angewandte Mathematik und Physik, 21, 192-218.

 Helmert, F. R. (1875). ~ber die Berechnung der wahrscheinlichen Fehlers aus einer endlichen Anzahl wahrer Beobachtungsfehler, Zeitschrift fur angewandte Mathematik und Physik, 20, 300-303.

 Wu, M. et al. (2019). Differentiable Antithetic Sampling for Variance Reduction in Stochastic Variational Inference. Online: https://arxiv.org/pdf/1810.02555.pdf

  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.