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 where these integrals are easy to calculate, 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 (a) the probability in any domain of a normal in any dimensions with any parameters; (b) the probability density, cumulative distribution, and inverse cumulative distribution of any function of a normal vector; (c) the classification errors among any number of normal distributions, the Bayes-optimal discriminability index, and relation to the receiver operating characteristic (ROC); (d) dimension reduction and visualizations for such problems; and (e) 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.

*unique*nonzero eigenvalues among \(D_i\), their degrees of freedom \({\boldsymbol k}\) are the numbers of times the eigenvalues occur and their noncentralities, and normal parameters are

*Bayes discriminability index*as the equal-variance index that corresponds to this same Bayes error, that is, the separation between two unit variance normals that have the same overlap as the two distributions, which comes out to be twice the \(z\)-score of the maximum accuracy:

*Biometrics,*24(3), 683–694. [CrossRef]

*Journal of Vision,*18(10), 549, https://doi.org/10.1167/18.10.549. [CrossRef]

*Biometrika,*60(2), 415–417. [CrossRef]

*Pattern classification*. Hoboken, New Jersey: John Wiley & Sons.

*Computation of multivariate normal and t probabilities*. Berlin/Heidelberg, Germany: Springer Science & Business Media.

*Journal of the Acoustical Society of America,*148(1), 222–225. [CrossRef]

*Signal detection theory and psychophysics*volume 1, (Vol. 1). New York: Wiley.

*Journal of Mathematical Psychology,*9(1), 128–139, www.sciencedirect.com/science/article/pii/0022249672900090.

*Vision Research,*43(23), 2451–2468. [CrossRef]

*Statistics & Probability Letters,*21(5), 417–419.

*I Annals of Mathematical Statistics,*31(3), 98–618.

*Annals of Mathematical Statistics,*33(2), 542–570.

*Psychological Bulletin,*80(6), 481.

*Approximation theory and approximation practice*. SIAM: Society for Industrial and Applied Mathematics, Philadelphia, PA.

*Journal of Vision,*20(13), 14, https://doi.org/10.1167/jov.20.13.14.

*Elementary signal detection theory*. Oxford, UK: Oxford University Press.