  A noncentral chi-square density is log-concave if the degree of freedom is
nu>=2. We complement this known result by showing that, for each 0<nu<2, there
exists lambda_nu>0 such that the chi-square with nu degrees of freedom and
noncentrality parameter lambda has a decreasing density if lambda <= lambda_nu,
and is bi-modal otherwise. The critical lambda_nu is characterized by an
equation involving a ratio of modified Bessel functions. When an interior mode
exists we derive precise bounds on its location.
