Het verbond tussen werkelijke verdeling, foutenverdeling en waarneem‐bare verdeling

The relation between the true distribution, the error distribution and the observable distribution. If ξ is a chance variable with a ‘true’ distribution function ξ (ξ), and if ϕ (x—ξ; ξ) represents the distribution of the errors made in observing ξ, than the actual observations will be distributed according to the ‘observable’ distribution which arises from a superposition of the true and the error distributions. By making use of moment generating functions, relations are derived between the fundamental parameters—mean, variance, β1 and β2—of the three distributions mentioned above, under the supposition that these functions are all continuous and differentiable, and that the error distribution is symmetric. When these assumptions do not hold similar methods may still be employed. In the general case that the error distribution depends on ξ we arrive at equations (6) to (12), which, if this distribution is independent of ξ, simplify to equations (13) to (16); if the error distribution is normal these can be further reduced to (19) to (21), a set of equations already established by Sittig1)in a previous paper. The same method of computation can be extended to cases of more than one dimension, and equations (23) to (27) express the relations between the variances and the correlation coefficients for a two‐dimensional distribution.