Power estimates of cuts of some improper integrals

Background. Gaussian distribution arises naturally in many applications and is widely used in a variety of theoretical constructs. The important role is played by a lower cut-off function $Q(x)$ of an improper integral from the density of a standard Gaussian distribution. The purpose of this paper is to obtain upper cuts for the arbitrary power of the function $Q(x)$ through the improper integral of the same type with a lower limit $ax$, where $a$ - an arbitrary parameter. Materials and methods. To obtain the necessary estimates the authors studied the behavior of the difference $Q^m(x)-Q(\sqrt{m}x)$ in various intervals of the real axis. At the same time, the well-known properties of the Gaussian distribution were widely used. In addition, the strict inequalities were brought to a special form of the exponential function, and upper and lower bounds for the function $Q(x)$ were obtained. Results. The paper shows that for any real $x$, when $m>2$, the inequality $Q^m(x)<Q(ax)$, where $a$ - an arbitrary number in the interval $[1;\sqrt{m}]$. In addition, it was found that this inequality cannot be improved on the parameter $a$. So, the paper shows, that the right border of the interval for $a$ can not be more than $\sqrt{m}$ and the left - can not be less than 1. Conclusions. The arbitrary degree function $Q(x)$ can be uniformly bounded above by a function of the same type with $ax$ argument. These estimates can be used in sociological and demographic studies in econometrics and statistics for point and interval estimates of the unknown parameters of the distribution.