  Univariate and multivariate normal probability distributions are widely used
when modeling decisions under uncertainty. Computing the performance of such
models requires integrating these distributions over specific domains, which
can vary widely across models. Besides some special cases, there exist no
general analytical expressions, standard numerical methods or software for
these integrals. Here we present mathematical results and open-source software
that provide (i) the probability in any domain of a normal in any dimensions
with any parameters, (ii) the probability density, cumulative distribution, and
inverse cumulative distribution of any function of a normal vector, (iii) the
classification errors among any number of normal distributions, the
Bayes-optimal discriminability index and relation to the operating
characteristic, (iv) ways to scale the discriminability of two distributions,
(v) dimension reduction and visualizations for such problems, and (vi) tests
for how reliably these methods may be used on given data. We demonstrate these
tools with vision research applications of detecting occluding objects in
natural scenes, and detecting camouflage.
