  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. %
