Positioning Error Probabilities for Some Forms of Center-of-Gravity Algorithm Calculated with the Cumulative Distributions. Part II

Abstract

To complete a previous work, the probability density functions for the errors in the center-of-gravity as positioning algorithm are derived with the usual methods of the cumulative distribution functions. These methods introduce substantial complications compared to the approaches used in a previous publication on similar problems. The combinations of random variables considered are: $X_{g3}=\theta (x_2-x_1)(x_1-x_3)/(x_1+x_2+x_3)+\theta (x_1-x_2)(x_1+2x_4)/(x_1+x_2+x_4)$ and $X_{g4}=(\theta (x_4-x_5)(2x_4+x_1-x_3)/(x_1+x_2+x_3+x_4)+\theta (x_5-x_4)(x_1-x_3-2x_5)/(x_1+x_2+x_3+x_5)$ The complete and partial forms of the probability density functions of these expressions of the center-of-gravity algorithms are calculated for general probability density functions of the observation noise. The cumulative probability distributions are the essential steps in this study, never calculated elsewhere.

16 pages, 11 figures. Another method to calculate the probabilities of arXiv:2011.14474