B. Ramachandran and C.R. Rao have proved that if X, X1, X2, . . ., Xn are i.i.d. random variables and if distributions of sample mean \bar X = \bar X(n) = (X1 + ··· + Xn)/n and monomial X are coincident at least at two points n = j1 and n = j2 such that log j1/ log j2 is irrational, then X follows a Cauchy law. Assuming that condition of coincidence of \bar X(n) and X are fulfilled at least for two n values, but only approximately, with some error ε in metric λ, we prove that, in certain sense, characteristic function of X is close to the characteristic function of the Cauchy distribution and construct stability estimation.

Effective interaction of students with information received in the lesson is the key to the success of mathematics teaching. Concrete methods in this direction are described in the article. The underlying thesis is the need for participation of the personality of teacher in this process.