Random processes such as temperature and acoustic noise are found in all types of mechanical systems. Knowledge of these processes can lead to improved design and detection methods related to faulty operation. The goal of this dissertation is to contribute to the knowledge base of such processes. Specifically, we address statistical signal processing methods that are appropriate and consistent relative to the physics of these systems. Two generic problems associated with random signal measurements from mechanical systems are addressed.;Random processes associated with mechanical systems usually have complex spectral structure containing both continuous and line spectral components. Accordingly, they are called mixed random processes. One problem addressed is to use variability related to families of spectral estimators for a mixed random process to better characterize its spectral information. We show that tones are a significant source of bias and variability of families of spectral estimators. Expressions for estimating statistical and arithmetic variability of three common families of spectral estimators are provided. An important and immediate application of these results is tone detection.;We also address the statistical problem of estimating the bandwidth parameter of a Gauss-Markov process from a realization of fixed and finite duration at selectable sampling interval. The motivation is that continuous-time processes are often sampled at a rate far higher than their underlying dynamics. It is commonly assumed a faster sample rate is better. But in many real world situations, such as in adaptive feedback control schemes design, short time changes demand only limited time being utilized. Thus this problem is investigated. The bias and variance expressions of the parameter estimator are derived with a second order expansion. Three sample rate regions---finite, large and very large ones, corresponding to substantial, gradual, and very slight variance drop, are quantitatively identified. Guidelines in choosing sampling rate based on estimator performance requirement are provided.;The results are used to characterize the stochastic structure of the sound pressure process from an engine cooling fan with and without mock engine, and to perform a hypothesis test for deciding whether a design change has a significant effect on the sound.