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 exists no general analytical expression, standard numerical method 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, distribution, and percentage points of any function of a normal vector, (iii) the error matrix that measures classification performance amongst any number of normal distributions, and the optimal discriminability index, (iv) dimension reduction and visualizations for such problems, and (v) tests for how reliably these methods can 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：在作出不确定的决定时,常态和多变的正常概率分布被广泛使用。计算这些模型的性能需要将这些分布在特定领域,这些分布在不同的模型中可能有很大差异。除了这些整体体易于计算的一些特殊案例外,对这些整体体没有一般的分析表达、标准数字方法或软件。这里我们展示数学结果和开放源软件,这些软件提供(一) 正常体任何层面的概率,并附有任何参数;(二) 正常矢量的任何函数的概率密度、分布和百分点;(三) 用于衡量任何正常分布体的分类性能的错误矩阵,以及最佳的可调和指数,(四) 这些问题的尺寸减少和可视化,以及(五) 关于如何可靠地在特定数据中使用这些方法的测试。我们用视觉研究应用来显示这些工具,在自然场景中探测occluting 对象,以及探测迷彩。