Let $X_i$ and $Y_i$ follow noncentral chi-square distributions with
the same degrees of freedom $\nu_i$ and noncentrality parameters $\delta_i^2$
and $\mu_i^2$, respectively, for $i = 1, \dots, n$, and let the $X_i$'s be
independent and the $Y_i$'s independent. A necessary and sufficient condition
is obtained under which $\Sigma_{i = 1}^n \lambda_i X_i$ is stochastically
smaller than $\Sigma_{i = 1}^n \lambda_i Y_i$ for all nonnegative real numbers
$\lambda_i \geq \dots \geq \lambda_n$. Reformulating this as a result in
geometric probability, solutions are obtained, in particular, to the problems
of monotonicity and location of extrema of the probability content of a rotated
ellipse under the standard bivariate Gaussian distribution. This complements
results obtained by Hall, Kanter and Perlman who considered the behavior of the
probability content of a square under rotation. More generally, it is shown
that the vector of partial sums $(X_1, X_1 + X_2, \dots, X_1 + \dots + X_n)$ is
stochastically smaller than $(Y_1, Y_1 + Y_2, \dots, Y_1 + \dots + Y_n)$ if and
only if $\Sigma_{i=1}^n \lambda_i X_i$ is stochastically smaller than
$\Sigma_{i=1}^n \lambda_i Y_i$ for all nonnegative real numbers $\lambda_1 \geq
\dots \geq \lambda_n$.