Дослідження розподілу випадкових хорд у сфері

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Дослідження розподілу випадкових хорд у сфері

Journal Title: Математичне моделювання - Year 2018, Vol 1, Issue 2

Abstract

STUDY OF THE DISTRIBUTION OF RANDOM CHORDS IN THE SPHERE Stroieva V.O, Avramenko V.I. Abstract In this paper we consider the problem of determining the length of a random chord, which is often associated with problems such as paradoxes. It is concluded that the use of three-dimensional models allows us to consider cases when uncertainty disappears when solving problems. It was investigated the distribution of random chords in a sphere by the example of rays from sources, whose positions may be different in relation to the sphere.If the source is sufficiently far from the middle of the sphere, then the rays can be considered parallel, and it turns out that the density distribution of the length of random chords is linear and expressed by the formula f(l)=l/(2r^2 ) 0<l<2r,where r is the radius of the sphere. For other provisions of the source relative to the sphere, the model of the uniform distribution in the distribution space is considered. It is assumed that this is a distribution, when the probability of attack on the surface element of the sphere of an arbitrary radius R with the center at the point of the source of the rays is constant To simulate such a distribution, an even distribution of each coordinate in the interval (-R, R) is used, using the condition that the distance from the point from the source does not exceed R.In this case, all three directions of the cosine of the rays are uniformly distributed, which ensures the same probability of impact on the element of the surface of the sphere.Separately obtained distribution laws for the position of the source both outside the scope and in the middle of the volume.In particular, in the position of the source on the surface of the sphere, the distribution of the lengths of random chords is uniform with the density of distribution f(l)= 0,5r 0 <l < 2r. Analyzing the influence of several random sources, it has been shown that mathematically it is described by the same dependencies as for one remote source. If the distribution of rays from a source is determined by a law that is not homogeneous, the distribution of the length of random chords in the sphere can be estimated by both analytical and statistical methods. References [1] Avramenko V.I. Probability Theoryand Mathematical Statistics: Teaching. manual. / V.I. Avramenko, I.K. Karimov – 2 ndform., рrocessing. and complemented – Dniprodzerzhinsk: DSTU, 2013. – 254 р. [2] Vukolov E.A. Collection of Math Problems for vtusov. Special courses. / E.A. Vukolov, A.V Efimov, V.N. Zemskov and others. Ed. A.VEfimova– М., 1984. – 607 р. [3] Mosteller F. Fifty entertaining probabilistic problems with solutions. / F. Mosteller. – M.: Nauka, 1975. – 112 p. [4] Sekey G. Paradoxes in probability theory and mathematical statistics. / G. Sekey. Per. from english–M.: Mir, 1990. – 240 p. [5] Stroieva V.O., Avramenko V. I. Investigation of the distribution of the length of the chord in the circle. / V.O. Stroeva, V.I. Avramenko // Questions of Applied Mathematics and Mathematical Modeling: a collection of scientific works. – D.: DNU, 2015. – W. 15, p. 181–190.

Authors and Affiliations

В. О. Строєва, В. І. Авраменко

Keywords

сфера хорда закон розподілу статистичне моделювання

EP ID EP444492
DOI 10.31319/2519-8106.2(39)2018.154201
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