Principles of Time Dependence of Impact Point Density
Saltanat Veysova1* 
, Sanan Amirov2
Abstract. The article considers the problem of determining possible values of impact points – a random variable and the probability of hitting a given interval based on the distribution law, presented in graphical and analytical form, as a result of a limited number of shots. By means of a mathematical analysis of the principle of the time dependence of the density of impact points during continuous shooting, images of the initial distribution functions and the density of the corresponding random variable were found. To study the distribution function of a random variable presented graphically, two distribution laws were considered: uniform and exponential, satisfying the basic conditions of the Laplace transform. The regularity of the analytical expression of the distribution density of a uniformly distributed random variable through its jumps in values corresponding to different moments in time is proven. Using this regularity, a distribution function was found that corresponds to jumps in density values. It was established that the analytical form of the distribution density, determined from the form of the fire scattering graph, is similar to the law of uniform distribution of a random variable, and its expression was established using the Heaviside function, and the Laplace transform was also found. It was shown that the Laplace transform of fire scattering, obeying a uniform and exponential law, more fully reflects the delay process at a certain point in time.
Keywords: random variable, distribution density, distribution function, regular and exponential distribution, Laplace transform, original, images, Heaviside function