Errors of Parameters of Distribution Functions for Spherical Bodies Stochastically Estimated with Parallel Test Lines of Regular and Narrow Intervals

Abstract

The parameters of some distribution functions for the radius of spherical bodies randomly dispersed in a three-dimensional space can be estimated with a grating of parallel test lines of regular intervals superposed on a test plane of unit surface area. The interval of the test lines is set narrower than the mean diameter of the circles on the test plane. The errors of number Nλo of chords per test line of unit length, arithmetical mean A and secondary moment (λ²) of the length of individual chords are: [C(Nλo) ^*]²=(4/9π)(Q₃/Q₂²)(1/M²λ, Nλo)+(4/π)(λ/Nλo), [C(λ)^*]²=(π/4)[(32/3π²)-1] (1/M²λNλo) + [(4/π)-(3π/8)(Q₂²/Q₃)](λ/Nλo) and [C(λ²)^*]₂=(1024/2025π)(Q₅Q₃/Q₄²) (1/M²λNλo)+4[(4/π)-(3π/8)(Q₂²/Q₃
)](λ/Nλo). The first and second terms of the right sides of these expressions represent intraregional and interregional errors, respectively; ^* denotes the error for the total region; M is the expectation of the number of test lines of unit length covering the test plane; and Q_n a quotient (Dⁿ)/Dⁿ of sphere diameter D, n being a positive integer. Intraregional errors are inversely proportional to M² with the grating, while they are inversely pro-portional only to M with random test lines. The use of regularly arranged test lines is consequently effective in minimizing intraregional errors.