Errors of Parameters of Distribution Functions for Spherical Bodies Stochastically Estimated on a Random Test Plane

Abstract

The parameters of some distribution functions for the radius of spherical bodies randomly dispersed in a three-dimensional space can be estimated on a random test plane of unit surface area. In this estimation the set of measured Nao, δ and (δ²) or that of Nλo, λ and (λ²) is used. They are number of circles, arithmetical mean and secondary moment of circle diameters or number of chords delivered by intersection of a test line of unit length with circles, arithmetical mean and secondary moment of chord length, respectively. Provided that the region of measurement is sufficiently large, intraregional errors of these quantities expressed as the squares of coefficient of variation C are approximately: [C(Nao)]w²=1/Nao, [C(δ)]w²=(32/3π²)(Q₃/Q₂²)(1/Nao) and [C(δ)]w²=(6/5)(Q₅/Q₃²)(1/Nao); [C(Nλo)]w²=1/MNλo, [C(λ)]w²=(9/8)(Q₄Q₂/Q₃²)(1/MNλo) and [C(λ)]w²=(4/3)(Q₆Q₂/Q₄²)(1/MNλo). In these expressions Q_n is a quotient defined by (Dⁿ)/Dⁿ, D and n being sphere diameter and a positive integer, respecitvely. The first three expressions may be used in these forms as the errors for the total region containing spheres. In the second three expressions M is the number of random test lines of unit length. When M is small, they can stand for the errors of the total region. In the case of large M, however, interregional errors or errors of sampling have to be added to them. The geometrical parameter of a distribution function of D is estimated from (δ²)/δ² or (λ²)/λ². The error of this ratio W is given by: [C(W)]²=[C(X²)-2C(X)]², where X stands for δ or λ.