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The performance of a univariate alarm system can be assessed in many cases by three indices, namely, the false alarm rate (FAR), missed alarm rate (MAR), and averaged alarm delay (AAD). First, this paper studies the definition and computation of the FAR, MAR, and AAD for the basic mechanism of alarm generation solely based on a trip point, and for the advanced mechanism of alarm generation by exploiting alarm on/off delays. Second, a systematic design of alarm systems is investigated based on the three performance indices and the tradeoffs among them. The computation of FAR, MAR, and AAD and the design of alarm systems require the probability density functions (PDFs) of the univariate process variable in the normal and abnormal conditions. Thus, a new method based on mean change detection is proposed to estimate the two PDFs. Numerical examples and an industrial case study are provided to validate the obtained theoretical results on the FAR, MAR and AAD, and to illustrate the proposed performance assessment and alarm system design procedures.