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) dimension reduction and visualizations for such problems, and (v) 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.

翻译：单变量与多变量正态概率分布广泛应用于不确定性下的决策建模。计算此类模型的性能需要对这些分布在特定区域（随模型不同而差异显著）进行积分。除若干特例外，这些积分尚无通用解析表达式、标准数值方法或现成软件支持。本文提出数学结果与开源软件，可提供：(i) 任意维度、任意参数的正态分布在任意区域内的概率；(ii) 正态随机向量任意函数的概率密度函数、累积分布函数及逆累积分布函数；(iii) 任意多个正态分布之间的分类误差、贝叶斯最优判别指数及其与操作特征的关系；(iv) 此类问题的降维与可视化方法；以及(v) 这些方法在给定数据上可靠性的检验。我们通过视觉研究中的自然场景遮挡物检测与伪装检测实例演示了这些工具。