The process of discretization of continuous distributions creates and provides a large set of discrete probabilistic models used in various statistical applications. The most common way of doing so is by considering the probability distribution of the integral part of a continuous random variable. In this note we explore the following problem related to the latter discretization process and pose the following question: If the family of distributions that is discretized is an exponential family on the real line, when the (integral) resulting discrete probability model also generates an exponential family? We give a complete answer to this question and provide necessary and sufficient conditions under which the discretized version of an exponential family is also an exponential family.

连续分布的离散化过程创建并提供了大量用于各种统计应用的离散概率模型。最常见的方法是考虑连续随机变量的积分部分的概率分布。在本笔记中，我们探讨了与后一种离散化过程相关的以下问题，并提出以下问题：如果离散化的分布族是实线上的指数族，则当（积分）所得离散概率模型也生成指数家庭？我们给出了这个问题的完整答案，并提供了指数族的离散化版本也是指数族的充要条件。