Some Power Integral Inequalities for the Gaussian Distribution
This work is devoted to obtaining power estimations of the function, where Ф(х) is the Laplace function. This difference is the lower cut of the integral of the standard Gaussian distribution density and shows the probability of getting a random variable distributed by the standard normal law into the interval (х, ∞). Because of the wide spread of applications and theoretical studies of the Gaussian distribution it is necessary to obtain estimates of the function. The peculiarity of the obtained results is that the power of the function Q(х) is evaluated not by the value of the integrand at some point, but through the values of the function at the point ax, where a is an arbitrary number from a closed interval. The difficulty of obtaining these estimates is that the density of the normal law is logarithmically concave function. In the paper we estimated the third and fourth degrees of the function Q(х). The theorem 1.1 shows that for and inequality Q3(х) < Q(aх) is correct. A similar result 1.2 is formulated in the theorem for the fourth degree of the function Q(х) and interval [1; 2]. It is important to note that the obtained estimates cannot be improved for the parameter a. Thus, the theorem 1.3 shows that the right border for the parameter a cannot be improved, and the theorem 1.4 shows that the left boundary cannot be less than 1.