An existence theorem for restrictions on the mean in the presence of a restriction on the dispersion

This article analyzes, from the purely mathematical point of view, a general practical problem. The problem consists in the influence of the scatter of experimental data on their mean values (and, possibly, on the probability) near the borders of intervals. The second central moment, the dispersion is a common measure of a scatter. Suppose, for instance, a nonnegative random variable X takes values in a finite interval . Write M for its mean. If there is a non-zero restriction on a central moment |E(X-M)n|≥|rnDisp.n|>0 under the condition 2≤n<∞, then A<(A+|rnDisp.n|/(B-A)n)≤M≤(B-|rnDisp.n|/(B-A)n). That is, |rnDisp.n|/(B-A)n)>0 is the width of a non-zero “forbidden zone” for the mean M near a border of the interval. Here, in the case of , this non-zero restriction is a restriction on the dispersion E(X-M)2≥r2Disp.2=σ2Min>0. So, if there is a non-zero restriction on the dispersion, then a non-zero “forbidden zone” exists for the mean near a border of the interval.