Probability Distributions of Positioning Errors for Some Forms of Center-of-Gravity Algorithms. Part II

The center of gravity is one of the most frequently used algorithm for position reconstruction with different analytical forms for the noise optimization. The error distributions of the different forms are essential instruments to improve the track fitting in particle physics. Their Cauchy-(Agnesi) tails have a beneficial effects to attenuate the outliers disturbance in the maximum likelihood search. The probability distributions are calculated for some combinations of random variables, impossible to find in literature, but relevant for track fitting: $x_{g3}=\theta(x_2-x_1)[ (x_1-x_3)/(x_1+x_2+x_3)] + \theta(x_1-x_2)[(x_1+2x_4)/(x_1+x_2+x_4)]$ and $x_{g4}=\theta(x_4-x_5)[(2x_4+x_1-x_3)/(x_1+x_2+x_3+x_4)]+ \theta(x_5-x_4)[(x_1-x_3-2x_5)/(x_1+x_2+x_3+x_5)]$ and $x_{g5}=(2x_4+x_1-x_3-2x_5)/(x_1+x_2+x_3+x_4+x_5)$. The probability density functions of $x_{g3}$, $x_{g4}$ and $x_{g5}$ have complex structures with regions of reduced probability. These regions must be handled with care to avoid false maximums in the likelihood function. General integral equations and detailed analytical expressions are calculated assuming the set $\{x_i\}$ as independent random variables with Gaussian probability distributions. %