Development of the convolutions of truncated normal random variables with three different quality characteristics in engineering applications

• We discuss why truncated distributions and their convolutions are critical in production processes.
• We develop those convolution models which have not been explored in the literature.
• The proposed convolution models will help estimate process yields more accurately.
• The proposed convolution models will also be useful in tolerance analysis.

In real-world situations, specifications are implemented to screen out nonconforming products as a part of screening inspections, which result in a truncated distribution for conforming products. Understanding these truncated probability density functions is paramount to the overall manufacturing industry, as more accurate evaluations of a process output will lead to a greater understanding of the process itself and associated costs. Furthermore, convolutions of truncated random variables play an important role in a multistage manufacturing system, where a screening inspection is performed at each stage. While the convolutions of normal distributions have been well established, the convolutions of truncated normal distributions have neither been understood clearly, nor have theoretical foundations been thoroughly explored in the literature, despite their practical importance. The mathematical framework and approximations using the error function for the convolutions of truncated normal random variables with three different types of quality characteristics are presented here. The convolutions established in this paper should enhance the accuracy and precision of real-world production processes particularly where components are required for assembling into the final product.