Positioning Error Probability for Some Forms of Center-of-Gravity Algorithms Calculated with the Cumulative Distributions. Part I

TL;DR

This paper derives cumulative distribution functions for center-of-gravity positioning algorithms using classical probability methods, focusing on complex combinations of independent random variables to assess error probabilities.

Contribution

It extends previous work by calculating cumulative distribution functions for specific center-of-gravity algorithms using classical methods, providing more detailed probabilistic analysis.

Findings

Derived cumulative distribution functions for key center-of-gravity algorithms.

Analyzed combinations of independent random variables relevant to track fitting.

Provided formulas useful for calculating positioning error probabilities.

Abstract

To complete a previous paper, the probability density functions of the center-of-gravity as positioning algorithm are derived with classical methods. These methods, as suggested by the textbook of Probability, require the preliminary calculation of the cumulative distribution functions. They are more complicated than those previously used for these tasks. In any case, the cumulative probability distributions could be useful. The combinations of random variables are those essential for track fitting $x={\xi}/{(\xi+\eta)}$, $x=\theta(x_3-x_1) (-x_3)/(x_3+x_2) +\theta(x_1-x_3)x_1/(x_1+x_2)$ and $x=(x_1-x_3)/(x_1+x_2+x_3)$. The first combination is a partial form of the two strip center-of-gravity. The second is the complete form, and the third is a simplified form of the three strip center-of-gravity. The cumulative probability distribution of the first expression was reported in the previous publications. The standard assumption is that $\xi$, $\eta$, $x_1$, $x_2$ and $x_3$ are independent random variables.