  In this paper we consider the probability density function (PDF) of the
non-central $\chi^2$ distribution with arbitrary number of degrees of freedom
and non-centrality. For this function we find the approximate location of the
maximum and discuss related edge cases of 1 and 2 degrees of freedom. We also
use this expression to demonstrate the improved performance of the C++ Boost's
implementation of the non-central $\chi^2$ and extend the domain of its
applicability.

Classical Analysis and ODEs