  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.
