POWER ESTIMATES OF CUTS OF SOME IMPROPER INTEGRALS

Pozhidaev Aleksandr Vasil'evich, Doctor of physical and mathematical sciences, professor, head of sub-department of higher mathematics, Siberian Transport University (191 D. Kovalchuk street, Novosibirsk, Russia), math@stu.ru
Pekel'nik Natal'ya Mikhaylovna, Candidate of pedagogical sciences, associate professor, sub-department of higher mathematics, Siberian Transport University (191 D. Kovalchuk street, Novosibirsk, Russia), pekelniknm@mail.ru
Khaustova Olesya Igorevna, Candidate of pedagogical sciences, associate professor, sub-department of higher mathematics, Siberian Transport University (191 D. Kovalchuk street, Novosibirsk, Russia), lex711@yandex.ru
Trefilova Irina Aleksandrovna, Lecturer, sub-department of higher mathematics, Siberian Transport University (191 D. Kovalchuk street, Novosibirsk, Russia), koja@mail.ru

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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 Qm(x) −Q( mx) 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 Qm(x) < Q(ax) , where a – an arbitrary number in the interval 1; m . In addition, it was found that this inequality can’t be improved on the parameter a . So, the paper shows, that the right border of the interval for a can not be more than 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.

probability density, gamma function, additional function of errors, logarithmically concave function, unimprovable values, Gaussian distribution, power estimations, distribution function.

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