This chapter is a collection of review material from signals and systems – continuous and discrete, and conversion between them – and from probability, random variable, and stochastic process.doi:10.1007/978-3-030-57706-3_9Sung-Moon Michael Yang...
In this section, we briefly review important concepts from probability theory. A Monte Carlo process is a sequence of random events. Often, a numerical outcome can be associated with each possible event. For example, when a fair die is thrown, the outcome could be any value from 1 to 6...
sample space: The set of all possible outcomes of a random process. discrete and continuous random process independent events logical independence For instance, two events occur but there is no reason to believe that two events affect one another. statistical independence Events and Conditional Probab...
Review of Random Variables and Random Processes Objectives Introduce some basic notions of probability. Introduce random variables and random processes. Introduce white noise and colored noise. Introduce frequency (spectral) decomposition of random processes. Introduction to Random Variables Most measurements ...
This work provides a comprehensive overview of the potentials and challenges of digital twins, laying a solid foundation for future research and advancements in this continually evolving field, in which statistics, probability and machine learning plays a crucial role....
The probability of failure can be determined using probabilistic methods, revealing valuable insights into decision-making processes. In contrast, deterministic approaches treat inputs such as material properties, climatic conditions, and geometry as discrete variables. However, these approaches may fail to...
most other threshold studies in the analysis (Fig.4). Importantly, Teneva et al.117also defined reef cells at risk as grid cells characterized by at least a 50% probability of experiencing 5-year mild or severe bleaching events by 2100....
The hyperbolic model of probability discounting functions21,22 has two free parameters: the parameter h, which reflects the discounting rate (i.e., the value of the probabilistic gain is discounted), and the parameter s, which governs the shape of the discounting function22. Figure 1 ...
2021; Sener and Feyzioglu 2022). A common assumption in applying most of SP methodologies is that the probability distributions of the random events are known or can be estimated with an acceptable accuracy. For majority of the SP problems, the objective is to identify a feasible solution that...
A stochastic process (or random process) is a probabilistic experiment or model that evolves in time. That is, each sample point (i.e., possible outcome) of the experiment is a function of time called a sample function. The sample space is the set of possible sample functions, and the ...