The purpose of this dissertation is to explore the uncertainty of process means and variances in order to improve processes with stochastic characteristics due to the complex nature of the underlying probability distributions. First, the gap between the existing conceptual notion of defects per million opportunities (DPMO) as part of process improvement initiatives and its applications to real-world engineering processes is explored. This is important because the current way of obtaining the DPMOs documented in the literature is problematic since it strictly assumes that there will be a shift in the process mean over time, while process variability remains unchanged. Accordingly, it does not account for shifting process standard deviation. This may not be the case in real-world practices. Several unique contributions to the Six Sigma body of knowledge are offered by expanding the existing DPMO and process fallout concepts, ultimately leading to process improvement. Second, convolutions of normal random variables are explored. Convolutions often arise in engineering problems, and the probability densities of the sums of these random variables are known in the literature. There are practical situations where specification limits on a process are imposed externally, and the product is typically scrapped if its performance does not fall in the specification range. The actual distribution after inspection is therefore truncated. Despite the practical importance of the role of truncated distributions, there has been little work on the theoretical foundation of convolutions associated with truncated random variables. This is paramount, since convolutions are often used as an important standard in statistical tolerance analysis. The convolutions of the combinations of truncated normal and truncated skew normal random variables on double and triple truncations are developed. This allows for a more accurate assessment of the mean and variance for a given process. Furthermore, it may not always be possible to define the specification limits and tolerances precisely within the limits of a probability density function for a process due to a relatively inaccurate or unstable process. This situation will be addressed through stochastic constrained programming
